ࡱ> /57./01234idfhpt v B D F HLgikmn|'` bjbjLULU 4.?.?%R\\\\8\ DC"0C0C0C+KK4KDFFFFFFAhF@MI+K@M@MF\\0C0CS[[[[@M\0C0CD[@MD[[z0C `6=0Mjdq0V3Sf3$3,L>OL,[{L$LLLLFF>[^LLL@M@M@M@M$$dd-$d-\\\\\\ Unit 7: Antiderivative & Integration DAYTOPICASSIGNMENT1Antiderivativesp.46-472Antiderivativesp.483Antiderivativesp.494Integration by Substitutionp.50-515Integration by Substitutionp.52-536Review for QuizWorksheet (Passed out in Class)7QUIZ 18Integrating Exponential Functionsp.549Integrating Exponential Functionsp.55-57 (Worksheet)10Integrating Rational Functionsp.5811Integrating Rational Functionsp.59-61 (Worksheet)12QUIZ 213Riemann Sums p.62-63 (Worksheet)14Riemann Sums p.64-65 (Worksheet)15Fundamental Theorem of Calculusp.6616Definite Integralsp.6717Definite Integralsp.68-71 (Worksheet)18Definite Integralsp.72-74 (Worksheet)19Review20TEST UNIT 7 7.1 Indefinite Integrals Calculus Learning Objectives A student will be able to: Find antiderivatives of functions. Represent antiderivatives. Interpret the constant of integration graphically. Solve differential equations. Use basic antidifferentiation techniques. Use basic integration rules. Introduction In this lesson we will introduce the idea of the antiderivative of a function and formalize as indefinite integrals. We will derive a set of rules that will aid our computations as we solve problems. Antiderivatives Definition A function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29" \* MERGEFORMATINET is called an antiderivative of a function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%28x%29%20%3D%20f%28x%29" \* MERGEFORMATINET for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET in the domain of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET  Example 1: Consider the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%203%20x%5E2." \* MERGEFORMATINET Can you think of a function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%28x%29%20%3D%20f%28x%29" \* MERGEFORMATINET ? (Answer:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29%20%3D%20x%5E3%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29%20%3D%20x%5E3%20-%206%2C" \* MERGEFORMATINET many other examples.) Since we differentiate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29" \* MERGEFORMATINET to get  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%2C" \* MERGEFORMATINET we see that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29%20%3D%20x%5E3%20%2B%20C" \* MERGEFORMATINET will work for any constant  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=C." \* MERGEFORMATINET Graphically, we can think the set of all antiderivatives as vertical transformations of the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29%20%3D%20x%5E3." \* MERGEFORMATINET The figure shows two such transformations.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123793" \* MERGEFORMATINET  With our definition and initial example, we now look to formalize the definition and develop some useful rules for computational purposes, and begin to see some applications. Notation and Introduction to Indefinite Integrals The process of finding antiderivatives is called antidifferentiation, more commonly referred to as integration. We have a particular sign and set of symbols we use to indicate integration:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20f%28x%29dx%20%3D%20F%28x%29%20%2B%20C." \* MERGEFORMATINET  We refer to the left side of the equation as the indefinite integral of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET with respect to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x." \* MERGEFORMATINET " The function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET is called the integrand and the constant  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=C" \* MERGEFORMATINET is called the constant of integration. Finally the symbol  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dx" \* MERGEFORMATINET indicates that we are to integrate with respect to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x." \* MERGEFORMATINET  Using this notation, we would summarize the last example as follows:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%203x%5E2%20dx%20%3D%20x%5E3%20%2B%20C" \* MERGEFORMATINET  Using Derivatives to Derive Basic Rules of Integration As with differentiation, there are several useful rules that we can derive to aid our computations as we solve problems. The first of these is a rule for integrating power functions,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5En%5C%20%5Bn%20%5Cneq%20-%201%5D%2C" \* MERGEFORMATINET and is stated as follows:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20x%5En%20dx%20%3D%20%5Cfrac%7B1%7D%20%7Bn%2B1%7D%20x%5E%7Bn%2B1%7D%20%2B%20C." \* MERGEFORMATINET  We can easily prove this rule. Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x%29%20%3D%20%5Cfrac%7B1%7D%20%7Bn%2B1%7D%20x%5E%7Bn%2B1%7D%20%2B%20C%2C%20n%20%5Cneq%20-1" \* MERGEFORMATINET . We differentiate with respect to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET and we have:  The rule holds for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5En" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Bn%20%5Cneq%20-1%5D." \* MERGEFORMATINET What happens in the case where we have a power function to integrate with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%20-1%2C" \* MERGEFORMATINET say  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20x%5E%7B-1%7D%20dx%20%3D%20%5Cint%20%5Cfrac%7B1%7D%20%7Bx%7D%20dx" \* MERGEFORMATINET . We can see that the rule does not work since it would result in division by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0" \* MERGEFORMATINET . However, if we pose the problem as finding  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%28x%29%20%3D%20%5Cfrac%7B1%7D%20%7Bx%7D" \* MERGEFORMATINET , we recall that the derivative of logarithm functions had this form. In particular,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cfrac%7Bd%7D%20%7Bdx%7D%20%5Cln%20x%20%3D%20%5Cfrac%7B1%7D%20%7Bx%7D" \* MERGEFORMATINET . Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20%5Cfrac%7B1%7D%20%7Bx%7D%20dx%20%3D%20%5Cln%20x%20%2B%20C." \* MERGEFORMATINET  In addition to logarithm functions, we recall that the basic exponentional function,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20e%5Ex%2C" \* MERGEFORMATINET was special in that its derivative was equal to itself. Hence we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20e%5Ex%20dx%20%3D%20e%5Ex%20%2B%20C." \* MERGEFORMATINET  Again we could easily prove this result by differentiating the right side of the equation above. The actual proof is left as an exercise to the student. As with differentiation, we can develop several rules for dealing with a finite number of integrable functions. They are stated as follows: If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g" \* MERGEFORMATINET are integrable functions, and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=C" \* MERGEFORMATINET is a constant, then  Example 2: Compute the following indefinite integral.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20%5Cleft%20%5B2x%5E3%20%2B%20%5Cfrac%7B3%7D%20%7Bx%5E2%7D%20-%20%5Cfrac%7B1%7D%20%7Bx%7D%5Cright%20%5D%20dx." \* MERGEFORMATINET  Solution: Using our rules we have  Sometimes our rules need to be modified slightly due to operations with constants as is the case in the following example. Example 3: Compute the following indefinite integral:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20e%5E%7B3x%7D%20dx." \* MERGEFORMATINET  Solution: We first note that our rule for integrating exponential functions does not work here since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cfrac%7Bd%7D%20%7Bdx%7D%20e%5E%7B3x%7D%20%3D%203e%5E%7B3x%7D." \* MERGEFORMATINET However, if we remember to divide the original function by the constant then we get the correct antiderivative and have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20e%5E%7B3x%7D%20dx%20%3D%20%5Cfrac%7Be%5E%7B3x%7D%7D%20%7B3%7D%20%2B%20C." \* MERGEFORMATINET  We can now re-state the rule in a more general form as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20e%5E%7Bkx%7D%20dx%20%3D%20%5Cfrac%7Be%5E%7Bkx%7D%7D%20%7Bk%7D%20%2B%20C." \* MERGEFORMATINET  Differential Equations We conclude this lesson with some observations about integration of functions. First, recall that the integration process allows us to start with function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET from which we find another function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%28x%29%20%3D%20f%28x%29." \* MERGEFORMATINET This latter equation is called a differential equation. This characterization of the basic situation for which integration applies gives rise to a set of equations that will be the focus of the Lesson on The Initial Value Problem. Example 4: Solve the general differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%28x%29%20%3D%20x%5E%5Cfrac%7B2%7D%20%7B3%7D%20%2B%20%5Csqrt%7Bx%7D." \* MERGEFORMATINET  Solution: We solve the equation by integrating the right side of the equation and have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20f%28x%29%20%3D%20%5Cint%20f%27%28x%29%20dx%20%3D%20%5Cint%20x%5E%7B%5Cfrac%7B2%7D%20%7B3%7D%7D%20dx%20%2B%20%5Cint%20%5Csqrt%7Bx%7D%20dx." \* MERGEFORMATINET  We can integrate both terms using the power rule, first noting that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Csqrt%7Bx%7D%20%3D%20x%5E%7B%5Cfrac%7B1%7D%20%7B2%7D%7D%2C" \* MERGEFORMATINET and have  Lesson Summary We learned to find antiderivatives of functions. We learned to represent antiderivatives. We interpreted constant of integration graphically. We solved general differential equations. We used basic antidifferentiation techniques to find integration rules. We used basic integration rules to solve problems. Multimedia Link The following applet shows a graph,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET and its derivative,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29" \* MERGEFORMATINET . This is similar to other applets we've explored with a function and its derivative graphed side-by-side, but this time  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET is on the right, and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29" \* MERGEFORMATINET is on the left. If you edit the definition of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29" \* MERGEFORMATINET , you will see the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET change as well. The  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET parameter adds a constant to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET . Notice that you can change the value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET without affecting  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29" \* MERGEFORMATINET . Why is this?  HYPERLINK "http://www.calculusapplets.com/antislope.html" Antiderivative Applet. Review Questions In problems #13, find an antiderivative of the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%201%20-%203%20x%5E2%20-%206x" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20x%20-%20x%5E%7B%5Cfrac%7B2%7D%20%7B3%7D%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Csqrt%5B5%5D%7B2x%20%2B%201%7D" \* MERGEFORMATINET  In #47, find the indefinite integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%282%20%2B%20%5Csqrt%7B5%7D%29%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%202%20%28x%20-%203%29%5E3%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%28x%5E2%20%5Ccdot%20%5Csqrt%5B3%5D%7Bx%7D%29%20dx" \* MERGEFORMATINET   EMBED Equation.DSMT4  Solve the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=f%27%28x%29%20%3D%204%20x%5E3%20-%203%20x%5E2%20%2B%20x%20-%203" \* MERGEFORMATINET . Find the antiderivative  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29" \* MERGEFORMATINET  of the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%202%20e%5E%7B2%20x%7D%20%2B%20x%20-%202" \* MERGEFORMATINET  that satisfies  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%20%280%29%20%3D%205." \* MERGEFORMATINET  Evaluate the indefinite integral  EMBED Equation.DSMT4 . (Hint: Examine the graph of EMBED Equation.DSMT4 .) Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29%20%3D%20x%20-%20x%5E3%20-%203%20x%5E2%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x%29%20%3D%20%5Cfrac%7Bx%5E2%7D%20%7B2%7D%20-%20%5Cfrac%7B3%7D%20%7B5%7D%20x%5E%7B%5Cfrac%7B5%7D%20%7B3%7D%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x%29%20%3D%20%5Cfrac%7B5%7D%20%7B12%7D%20%282x%20%2B%201%29%5E%7B%5Cfrac%7B6%7D%20%7B5%7D%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%5Cleft%20%282%20%2B%20%5Csqrt%7B5%7D%5Cright%20%29%20dx%20%3D%202x%20%2B%20%5Csqrt%7B5%7Dx%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%202%28x%20-3%29%5E3%20dx%20%3D%20%5Cfrac%7B%28x%20-%203%29%5E4%7D%20%7B2%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%5Cleft%20%28x%5E2%20%5Ccdot%20%5Csqrt%5B3%5D%7Bx%7D%5Cright%20%29%20dx%20%3D%20%5Cfrac%7B3%7D%20%7B10%7D%20x%5E%7B%5Cfrac%7B10%7D%20%7B3%7D%7D%20%2B%20C" \* MERGEFORMATINET   EMBED Equation.DSMT4   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20x%5E4%20-%20x%5E3%20%2B%20%5Cfrac%7Bx%5E2%7D%20%7B2%7D%20-%203x%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x%29%20%3D%20e%5E%7B2x%7D%20%2B%20%5Cfrac%7Bx%5E2%7D%20%7Bx%7D%20-%202x%20%2B%204" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%7Cx%7C%20dx%20%3D%20%5Cfrac%7Bx%5E2%7D%20%7B2%7D%20%2B%20C" \* MERGEFORMATINET  Indefinite Integrals Practice Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  Integrate. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  h.  EMBED Equation.DSMT4  i.  EMBED Equation.DSMT4  j.  EMBED Equation.DSMT4  k.  EMBED Equation.DSMT4  l.  EMBED Equation.DSMT4  m.  EMBED Equation.DSMT4  n.  EMBED Equation.DSMT4  o.  EMBED Equation.DSMT4  p.  EMBED Equation.DSMT4  q.  EMBED Equation.DSMT4  3. Find two functions that have the given derivative and sketch the graph of each. a.  INCLUDEPICTURE "C:\\Program Files\\TI Education\\TI InterActive!\\TIIimagefile13560.gif" \* MERGEFORMATINET  b.  INCLUDEPICTURE "C:\\Program Files\\TI Education\\TI InterActive!\\TIIimagefile13919.gif" \* MERGEFORMATINET   Answers: (Of course, you could have checked all of yours using differentiation!) 2. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  h.  EMBED Equation.DSMT4  i.  EMBED Equation.DSMT4  j.  EMBED Equation.DSMT4  k.  EMBED Equation.DSMT4  l.  EMBED Equation.DSMT4  m.  EMBED Equation.DSMT4  n.  EMBED Equation.DSMT4  o.  EMBED Equation.DSMT4  p.  EMBED Equation.DSMT4  q.  EMBED Equation.DSMT4  7.2 The Initial Value Problem Learning Objectives Find general solutions of differential equations Use initial conditions to find particular solutions of differential equations Introduction In the Lesson on Indefinite Integrals Calculus we discussed how finding antiderivatives can be thought of as finding solutions to differential equations:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%28x%29%20%3D%20f%28x%29." \* MERGEFORMATINET We now look to extend this discussion by looking at how we can designate and find particular solutions to differential equations. Lets recall that a general differential equation will have an infinite number of solutions. We will look at one such equation and see how we can impose conditions that will specify exactly one particular solution. Example 1: Suppose we wish to solve the following equation:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20f%27%28x%29%20%3D%20e%5E%7B3x%7D%20-%206%5Csqrt%7Bx%7D." \* MERGEFORMATINET  Solution: We can solve the equation by integration and we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20f%28x%29%20%3D%5Cfrac%7B1%7D%20%7B3%7D%20e%5E%7B3x%7D%20-%204x%5E%7B%5Cfrac%7B3%7D%20%7B2%7D%7D%20%2B%20C." \* MERGEFORMATINET  We note that there are an infinite number of solutions. In some applications, we would like to designate exactly one solution. In order to do so, we need to impose a condition on the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET We can do this by specifying the value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET for a particular value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x." \* MERGEFORMATINET In this problem, suppose that add the condition that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%280%29%20%3D%201." \* MERGEFORMATINET This will specify exactly one value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=C" \* MERGEFORMATINET and thus one particular solution of the original equation: Substituting  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%20%280%29%20%3D%201" \* MERGEFORMATINET into our general solution  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7B1%7D%20%7B3%7D%20e%5E%7B3x%7D%20-%204x%5E%7B%5Cfrac%7B3%7D%20%7B2%7D%7D%20%2B%20C" \* MERGEFORMATINET gives  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%201%20%3D%20%5Cfrac%7B1%7D%20%7B3%7D%20e%5E%7B3%280%29%7D%20-%204%280%29%5E%7B%5Cfrac%7B3%7D%20%7B2%7D%7D%20%2B%20C" \* MERGEFORMATINET or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20C%20%3D%201%20-%20%5Cfrac%7B1%7D%20%7B3%7D%20%3D%20%5Cfrac%7B2%7D%20%7B3%7D." \* MERGEFORMATINET Hence the solution  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7B1%7D%20%7B3%7D%20e%5E%7B3x%7D%20-%204x%5E%7B%5Cfrac%7B3%7D%20%7B2%7D%7D%20%2B%20%5Cfrac%7B2%7D%20%7B3%7D" \* MERGEFORMATINET is the particular solution of the original equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%28x%29%20%3D%20e%5E%7B3x%7D%20-%206%5Csqrt%7Bx%7D" \* MERGEFORMATINET satisfying the initial condition  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%20%280%29%20%3D%201." \* MERGEFORMATINET  We now can think of other problems that can be stated as differential equations with initial conditions. Consider the following example. Example 2: Suppose the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET includes the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%282%20%2C6%29" \* MERGEFORMATINET and that the slope of the tangent line to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET at any point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET is given by the expression  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=3%20x%20%2B%204." \* MERGEFORMATINET Find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%20%28-2%29." \* MERGEFORMATINET  Solution: We can re-state the problem in terms of a differential equation that satisfies an initial condition.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%203%20x%20%2B%204" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%20%282%29%20%3D%206" \* MERGEFORMATINET . By integrating the right side of the differential equation we have  Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7B3%7D%20%7B2%7D%20x%5E2%20%2B%204x%20-%208" \* MERGEFORMATINET is the particular solution of the original equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%203%20x%20%2B%204" \* MERGEFORMATINET satisfying the initial condition  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%20%282%29%20%3D%206." \* MERGEFORMATINET  Finally, since we are interested in the value  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%20%28-2%29" \* MERGEFORMATINET , we put  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-2" \* MERGEFORMATINET into our expression for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET and obtain:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=f%28-2%29%3D-10" \* MERGEFORMATINET  Lesson Summary We found general solutions of differential equations. We used initial conditions to find particular solutions of differential equations. Multimedia Link The following applet allows you to set the initial equation for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29" \* MERGEFORMATINET and then the slope field for that equation is displayed. In magenta you'll see one possible solution for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET . If you move the magenta point to the initial value, then you will see the graph of the solution to the initial value problem. Follow the directions on the page with the applet to explore this idea, and then try redoing the examples from this section on the applet.  HYPERLINK "http://www.calculusapplets.com/slopefields.html" Slope Fields Applet. Review Questions In problems #13, solve the differential equation for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%20%28x%29." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%28x%29%20%3D%202e%5E%7B2x%7D%20-%202%5Csqrt%7Bx%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%28x%29%20%3D%20%5Csin%20x%20-%20%5Cfrac%7B1%7D%20%7Be%5Ex%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%27%28x%29%20%3D%20%282%20%2B%20x%29%20%5Csqrt%7Bx%7D" \* MERGEFORMATINET  In problems #47, solve the differential equation for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29" \* MERGEFORMATINET  given the initial condition.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%28x%29%20%3D%206x%5E5%20-%204x%5E2%20%2B%20%5Cfrac%7B7%7D%20%7B3%7D" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%281%29%20%3D%204" \* MERGEFORMATINET .  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%27%28x%29%20%3D%203%20x%5E2%20%2B%20e%5E%7B2%20x%7D" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%20%280%29%20%3D%203." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%28x%29%20%3D%20%5Csqrt%5B3%5D%7Bx%5E2%7D%20-%20%5Cfrac%7B1%7D%20%7Bx%5E2%7D" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%281%29%20%3D%203" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%28x%29%20%3D%20%282%5Ccos%20x%20-%20%5Csin%20x%29%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20-%20%5Cfrac%7B%5Cpi%7D%20%7B2%7D%20%5Cle%20x%20%5Cle%20%5Cfrac%7B%5Cpi%7D%20%7B2%7D%2C" \* MERGEFORMATINET  and  EMBED Equation.DSMT4  Suppose the graph of f includes the point (-2, 4) and that the slope of the tangent line to f at x is -2x+4. Find f(5). In problems #910, find the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET that satisfies the given conditions.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%27%28x%29%20%3D%20%5Csin%20x%20-%20e%5E%7B-2x%7D" \* MERGEFORMATINET  with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%280%29%20%3D%20%5Cfrac%7B5%7D%20%7B2%7D" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%280%29%20%3D%200" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%27%27%28x%29%20%3D%20%5Cfrac%7B1%7D%20%7B%5Csqrt%7Bx%7D%7D" \* MERGEFORMATINET  with  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%202%5E%7B2x%7D%20-%20%5Cfrac%7B4%7D%20%7B3%7D%20x%5E%7B%5Cfrac%7B3%7D%20%7B2%7D%7D%20%2B%20C%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20-%5Ccos%20x%20%2B%20%5Cfrac%7B1%7D%20%7Be%5Ex%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7B8%7D%20%7B15%7D%20x%5E%7B%5Cfrac%7B5%7D%20%7B2%7D%7D%20%2B%20%5Cfrac%7B4%7D%20%7B35%7D%20x%5E%7B%5Cfrac%7B7%7D%20%7B2%7D%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20x%5E6%20-%20%5Cfrac%7B4%7D%20%7B3%7D%20x%5E3%20%2B%20%5Cfrac%7B7%7D%20%7B3%7D%20x%20%2B%202" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20x%5E3%20%2B%20%5Cfrac%7Be%5E%7B2x%7D%7D%20%7B2%7D%20%2B%20%5Cfrac%7B5%7D%20%7B2%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7B3%7D%20%7B5%7D%20%5Csqrt%5B3%5D%7Bx%5E5%7D%20%2B%20%5Cfrac%7B1%7D%20%7Bx%7D%20%2B%20%5Cfrac%7B7%7D%20%7B5%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%202%5Csin%20x%20%2B%20%5Ccos%20x%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20-x%5E2%20%2B%204%20x%20%2B%2016" \* MERGEFORMATINET ;  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%20%285%29%20%3D%2011" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20-%5Csin%20x%20-%20%5Cfrac%7B1%7D%20%7B4%7D%20e%5E%7B2x%7D%20%2B%204x%20%2B%20%5Cfrac%7B1%7D%20%7B4%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7B4%7D%20%7B3%7D%20x%5E%7B%5Cfrac%7B3%7D%20%7B2%7D%7D%20%2B%203x%20%2B%20%5Cfrac%7B7%7D%20%7B3%7D" \* MERGEFORMATINET  Initial Condition & Integration of Trig Functions Practice Find the particular solution  EMBED Equation.DSMT4  that satisfies the differential equation and initial condition. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  Find the equation of the function  EMBED Equation.DSMT4  whose graph passes through the point.  EMBED Equation.DSMT4  Find the function  EMBED Equation.DSMT4  that satisfies the given conditions. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  Integrate. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  Answers: 1. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  4. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  7.3 The Area Problem Learning Objectives Use sigma notation to evaluate sums of rectangular areas Find limits of upper and lower sums Use the limit definition of area to solve problems Introduction In The Lesson The Calculus we introduced the area problem that we consider in integral calculus. The basic problem was this:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2." \* MERGEFORMATINET Suppose we are interested in finding the area between the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x-" \* MERGEFORMATINET axis and the curve of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%2C" \* MERGEFORMATINET from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123774" \* MERGEFORMATINET  We approximated the area by constructing four rectangles, with the height of each rectangle equal to the maximum value of the function in the sub-interval.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123791" \* MERGEFORMATINET  We then summed the areas of the rectangles as follows: and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%20R_1%20%2B%20R_2%20%2B%20R_3%20%2B%20R_4%20%3D%20%5Cfrac%7B30%7D%20%7B64%7D%20%3D%20%5Cfrac%7B15%7D%20%7B32%7D%20%5Capprox%200.46." \* MERGEFORMATINET  We call this the upper sum since it is based on taking the maximum value of the function within each sub-interval. We noted that as we used more rectangles, our area approximation became more accurate. We would like to formalize this approach for both upper and lower sums. First we note that the lower sums of the area of the rectangles results in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=R_1%20%2B%20R_2%20%2B%20R_3%20%2B%20R_4%20%3D%2013/64%20%5Capprox%200.20" \* MERGEFORMATINET Our intuition tells us that the true area lies somewhere between these two sums, or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0.20%20%3C%20%5C%3B%5Cmathrm%7BArea%7D%5C%20%3C%200.46" \* MERGEFORMATINET and that we will get closer to it by using more and more rectangles in our approximation scheme. In order to formalize the use of sums to compute areas, we will need some additional notation and terminology. Sigma Notation In The Lesson The Calculus we used a notation to indicate the upper sum when we increased our rectangles to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20N%20%3D%2016" \* MERGEFORMATINET and found that our approximation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%20%3D%20%5Csum_1%5E%7B16%7D%20R_i%20%3D%20%5Cfrac%7B195%7D%20%7B512%7D%20%5Capprox%200.38" \* MERGEFORMATINET . The notation we used to enabled us to indicate the sum without the need to write out all of the individual terms. We will make use of this notation as we develop more formal definitions of the area under the curve. Lets be more precise with the notation. For example, the quantity  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20A%20%3D%20%5Csum%20R_i" \* MERGEFORMATINET was found by summing the areas of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=N%20%3D%2016" \* MERGEFORMATINET rectangles. We want to indicate this process, and we can do so by providing indices to the symbols used as follows:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20A%20%3D%20%5Csum_%7Bi%3D1%7D%5E%7B16%7D%20R_i%20%3D%20R_1%20%2B%20R_2%20%2B%20R_3%20%2B%20.%20.%20.%20%2B%20R_%7B15%7D%20%2B%20R_%7B16%7D." \* MERGEFORMATINET  The sigma symbol with these indices tells us how the rectangles are labeled and how many terms are in the sum. Useful Summation Formulas We can use the notation to indicate useful formulas that we will have occasion to use. For example, you may recall that the sum of the first  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET integers is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%28%20n%20%2B%201%29/2." \* MERGEFORMATINET We can indicate this formula using sigma notation. The formula is given here along with two other formulas that will become useful to us.  We can show from associative, commutative, and distributive laws for real numbers that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csum_%7Bi%20%3D%201%7D%5En%20%28a_i%20%2B%20b_i%29%20%3D%20%5Csum_%7Bi%20%3D%201%7D%5En%20%28a_i%29%20%2B%20%5Csum_%7Bi%20%3D%201%7D%5En%20%28b_i%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csum_%7Bi%20%3D%201%7D%5En%20%28k%20a_i%29%20%3D%20k%20%5Csum_%7Bi%20%3D%201%7D%5En%20%28a_i%29." \* MERGEFORMATINET  Example 1: Compute the following quantity using the summation formulas:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Csum_%7Bi%20%3D%201%7D%5E%7B10%7D%202i%28i%20-%206i%29." \* MERGEFORMATINET  Solution:  Another Look at Upper and Lower Sums We are now ready to formalize our initial ideas about upper and lower sums. Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET be a bounded function in a closed interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%20%3D%20%5Bx_0%2C%20%5Cldots%2C%20x_n%5D" \* MERGEFORMATINET the partition of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%20%2C%20b%5D" \* MERGEFORMATINET into  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET subintervals. We can then define the lower and upper sums, respectively, over partition  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P" \* MERGEFORMATINET , by  where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m_i" \* MERGEFORMATINET is the minimum value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET in the interval of length  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_i%20-%20x_%7Bi%20-%201%7D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=M_i" \* MERGEFORMATINET is the maximum value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET in the interval of length  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_i%20-%20x_%7Bi%20-%201%7D." \* MERGEFORMATINET  The following example shows how we can use these to find the area. Example 2: Show that the upper and lower sums for the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%2C" \* MERGEFORMATINET from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201%2C" \* MERGEFORMATINET approach the value  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%20%3D%201/3." \* MERGEFORMATINET  Solution: Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P" \* MERGEFORMATINET be a partition of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET equal sub intervals over  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C%201%5D." \* MERGEFORMATINET We will show the result for the upper sums. By our definition we have  We note that each rectangle will have width  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cfrac%7B1%7D%20%7Bn%7D%2C" \* MERGEFORMATINET   indicated:  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123781" \* MERGEFORMATINET   We can re-write this result as:  We observe that as  We now are able to define the area under a curve as a limit. Definition Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET be a continuous function on a closed interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%20%2Cb%5D." \* MERGEFORMATINET Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P" \* MERGEFORMATINET be a partition of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET equal sub intervals over  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%20%2Cb%5D." \* MERGEFORMATINET Then the area under the curve of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is the limit of the upper and lower sums, that is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20A%20%3D%20%5Clim_%7Bn%20%5Crightarrow%20%2B%5Cinfty%7D%20S%28P%29%20%3D%20%5Clim_%7Bn%20%5Crightarrow%20%2B%5Cinfty%7D%20T%28P%29." \* MERGEFORMATINET  Example 3: Use the limit definition of area to find the area under the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%204%20-%20x" \* MERGEFORMATINET from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=1" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123788" \* MERGEFORMATINET  Solution: If we partition the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B1%20%2C3%5D" \* MERGEFORMATINET into  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET equal sub-intervals, then each sub-interval will have length  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cfrac%7B3%20-%201%7D%20%7Bn%7D%20%3D%20%5Cfrac%7B2%7D%20%7Bn%7D" \* MERGEFORMATINET and height  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=3%20-%20i%20%5Ctriangle%20x" \* MERGEFORMATINET as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=i" \* MERGEFORMATINET varies from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=1" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n." \* MERGEFORMATINET So we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Ctriangle%20x%20%3D%20%5Cfrac%7B2%7D%20%7Bn%7D" \* MERGEFORMATINET and  Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Ctriangle%20x%20%3D%20%5Cfrac%7B2%7D%20%7Bn%7D" \* MERGEFORMATINET , we then have by substitution as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20n%20%5Crightarrow%20%5Cinfty" \* MERGEFORMATINET . Hence the area is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%20%3D%204." \* MERGEFORMATINET  This example may also be solved with simple geometry. It is left to the reader to confirm that the two methods yield the same area. Lesson Summary We used sigma notation to evaluate sums of rectangular areas. We found limits of upper and lower sums. We used the limit definition of area to solve problems. Review Questions In problems #12 , find the summations.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Csum_%7Bi%20%3D%201%7D%5E%7B10%7D%20i%282i%20-%203%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Csum_%7Bi%20%3D%201%7D%5En%20%283%20-%20i%29%20%282%20%2B%20i%29" \* MERGEFORMATINET  In problems #35, find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=S%28P%29" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=T%28P%29" \* MERGEFORMATINET  under the partition P.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%201%20-%20x%5E2%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%20%3D%20%5Cleft%20%5C%7B%200%2C%20%5Cfrac%7B1%7D%20%7B2%7D%2C%201%2C%20%5Cfrac%7B3%7D%20%7B2%7D%2C%202%20%5Cright%20%5C%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%202x%5E2%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%20%3D%20%5Cleft%20%5C%7B%20-1%2C%20-%5Cfrac%7B1%7D%20%7B2%7D%2C%200%2C%20%5Cfrac%7B1%7D%20%7B2%7D%2C%201%20%5Cright%20%5C%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7B1%7D%20%7Bx%7D%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%20%3D%20%5Cleft%20%5C%7B%20-4%2C%20-3%2C%20-2%2C%20-1%20%5Cright%20%5C%7D" \* MERGEFORMATINET  In problems #68, find the area under the curve using the limit definition of area.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%203%20x%20%2B%205" \* MERGEFORMATINET  from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202" \* MERGEFORMATINET  to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%206." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2" \* MERGEFORMATINET  from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201" \* MERGEFORMATINET  to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7B1%7D%20%7Bx%7D" \* MERGEFORMATINET  from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201" \* MERGEFORMATINET  to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%204." \* MERGEFORMATINET  In problems #910, state whether the function is integrable in the given interval. Give a reason for your answer.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%7Cx%20-%202%7C" \* MERGEFORMATINET  on the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B1%20%2C4%5D" \* MERGEFORMATINET   on the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%20%2C1%5D" \* MERGEFORMATINET  Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Csum_%7Bi%20%3D%201%7D%5E%7B10%7D%20i%282i%20-%203%29%20%3D%20605" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Csum_%7Bi%20%3D%201%7D%5En%20%283%20-%20i%29%20%282%20%2B%20i%29%20%3D%20%5Cfrac%7B1%7D%20%7B3%7D%20%2819%20-%20n%5E2%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=S%20%28P%29%20%3D%20-1.75" \* MERGEFORMATINET ;  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=T%28P%29%20%3D%200.25" \* MERGEFORMATINET  (note that we have included areas under the x-axis as negative values.)  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=S%20%28P%29%20%3D%200.5" \* MERGEFORMATINET ;  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=T%28P%29%20%3D%202.5" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=S%20%28P%29%20%3D%20-1.83" \* MERGEFORMATINET ;  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=T%28P%29%20%3D%20-1.08" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%2068" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%20%5Cfrac%7B26%7D%20%7B3%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%20%5Cfrac%7B15%7D%20%7B16%7D" \* MERGEFORMATINET  Yes, since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%7Cx%20-%202%7C" \* MERGEFORMATINET  is continuous on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B1%2C%204%5D" \* MERGEFORMATINET  No, since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=S%28P%29%20%3D%20-1" \* MERGEFORMATINET ;  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=T%28P%29%20%3D%201" \* MERGEFORMATINET  7.4 Definite Integrals Learning Objectives Use Riemann Sums to approximate areas under curves Evaluate definite integrals as limits of Riemann Sums Introduction In the Lesson The Area Problem we defined the area under a curve in terms of a limit of sums.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20A%20%3D%20%5Clim_%7Bn%20%5Cto%20%2B%5Cinfty%7D%20S%28P%29%20%3D%20%5Clim_%7Bn%20%5Cto%20%2B%5Cinfty%7D%20T%28P%29" \* MERGEFORMATINET  where   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=S%28P%29%2C" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=T%28P%29" \* MERGEFORMATINET were examples of Riemann Sums. In general, Riemann Sums are of form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Csum_%7Bi%20%3D%201%7D%5En%20f%28x_i%5E%2A%29%20%5Ctriangle%20x" \* MERGEFORMATINET where each  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_i%5E%2A" \* MERGEFORMATINET is the value we use to find the length of the rectangle in the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=i%5E%7Bth%7D" \* MERGEFORMATINET sub-interval. For example, we used the maximum function value in each sub-interval to find the upper sums and the minimum function in each sub-interval to find the lower sums. But since the function is continuous, we could have used any points within the sub-intervals to find the limit. Hence we can define the most general situation as follows: Definition If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is continuous on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D%2C" \* MERGEFORMATINET we divide the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET into  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET sub-intervals of equal width with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Ctriangle%20x%20%3D%20%5Cfrac%7Bb%20-%20a%7D%20%7Bn%7D" \* MERGEFORMATINET . We let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_0%20%3D%20a%2C%20x_1%2C%20x_2%2C%20%5Cldots%2C%20x_n%20%3D%20b" \* MERGEFORMATINET be the endpoints of these sub-intervals and let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_1%5E%2A%2C%20x_2%5E%2A%2C%20%5Cldots%2C%20x_n%5E%2A" \* MERGEFORMATINET be any sample points in these sub-intervals. Then the definite integral of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20a" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20b" \* MERGEFORMATINET is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint_a%5Eb%20f%28x%29%20dx%20%3D%20%5Clim_%7Bn%20%5Crightarrow%20%5Cinfty%7D%20%5Csum_%7Bi%20%3D%201%7D%5En%20f%28x_i%5E%2A%29%20%5Ctriangle%20x." \* MERGEFORMATINET  Example 1: Evaluate the Riemann Sum for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3" \* MERGEFORMATINET from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203" \* MERGEFORMATINET using  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%206" \* MERGEFORMATINET sub-intervals and taking the sample points to be the midpoints of the sub-intervals. Solution: If we partition the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C%203%5D" \* MERGEFORMATINET into  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%206" \* MERGEFORMATINET equal sub-intervals, then each sub-interval will have length  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cfrac%7B3%20-%200%7D%20%7B6%7D%20%3D%20%5Cfrac%7B1%7D%20%7B2%7D." \* MERGEFORMATINET So we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Ctriangle%20x%20%3D%5Cfrac%7B1%7D%20%7B2%7D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123787" \* MERGEFORMATINET   Now lets compute the definite integral using our definition and also some of our summation formulas. Example 2: Use the definition of the definite integral to evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E3%20x%5E3%20dx." \* MERGEFORMATINET  Solution: Applying our definition, we need to find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint_0%5E3%20x%5E3%20dx%20%3D%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Csum_%7Bi%20%3D%201%7D%5En%20f%28x_i%5E%2A%29%20%5Ctriangle%20x." \* MERGEFORMATINET  We will use right endpoints to compute the integral. We first need to divide  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C%203%5D" \* MERGEFORMATINET into  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET sub-intervals of length  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Ctriangle%20x%20%3D%20%5Cfrac%7B3%20-%200%7D%20%7Bn%7D%20%3D%20%5Cfrac%7B3%7D%20%7Bn%7D." \* MERGEFORMATINET Since we are using right endpoints,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x_0%20%3D%200%2C%20x_1%20%3D%20%5Cfrac%7B3%7D%20%7Bn%7D%2C%20x_2%20%3D%20%5Cfrac%7B6%7D%20%7Bn%7D%2C%20%5Cldots%20x_i%20%3D%20%5Cfrac%7B3i%7D%20%7Bn%7D." \* MERGEFORMATINET  So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E3%20x%5E3%20dx%20%3D" \* MERGEFORMATINET  Recall that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Csum_1%5En%20i%5E3%20%3D%20%5Cleft%20%5B%5Cfrac%7Bn%28n%20%2B1%29%7D%20%7B2%7D%20%5Cright%5D%5E2." \* MERGEFORMATINET By substitution, we have  Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint_0%5E3%20x%5E3%20dx%20%3D%20%5Cfrac%7B81%7D%20%7B4%7D." \* MERGEFORMATINET  Before we look to try some problems, lets make a couple of observations. First, we will soon not need to rely on the summation formula and Riemann Sums for actual computation of definite integrals. We will develop several computational strategies in order to solve a variety of problems that come up. Second, the idea of definite integrals as approximating the area under a curve can be a bit confusing since we may sometimes get results that do not make sense when interpreted as areas. For example, if we were to compute the definite integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_%7B-3%7D%5E3%20x%5E3%20dx%2C" \* MERGEFORMATINET then due to the symmetry of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3" \* MERGEFORMATINET about the origin, we would find that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_%7B-3%7D%5E3%20x%5E3%20dx%20%3D%200." \* MERGEFORMATINET This is because for every sample point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_j%5E%2A%2C" \* MERGEFORMATINET we also have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-x_j%5E%2A" \* MERGEFORMATINET is also a sample point with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28-%20x_j%5E%2A%29%20%3D%20-%20f%28x_j%5E%2A%29." \* MERGEFORMATINET Hence, it is more accurate to say that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_%7B-3%7D%5E3%20x%5E3%20dx" \* MERGEFORMATINET gives us the net area between  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-3" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203." \* MERGEFORMATINET If we wanted the total area bounded by the graph and the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x-" \* MERGEFORMATINET axis, then we would compute  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%202%20%5Cint_0%5E3%20x%5E3%20dx%20%3D%20%5Cfrac%7B81%7D%20%7B2%7D" \* MERGEFORMATINET . Lesson Summary We used Riemann Sums to approximate areas under curves. We evaluated definite integrals as limits of Riemann Sums. Multimedia Link For video presentations on calculating definite integrals using Riemann Sums (13.0), see  HYPERLINK "http://www.youtube.com/watch?v=gFpHHTxsDkI" Riemann Sums, Part 1 (6:15) INCLUDEPICTURE "http://i4.ytimg.com/vi/gFpHHTxsDkI/hqdefault.jpg" \* MERGEFORMATINET  and  HYPERLINK "http://www.youtube.com/watch?v=GE4OLfmJ8P8" Riemann Sums, Part 2 (8:32) INCLUDEPICTURE "http://i4.ytimg.com/vi/GE4OLfmJ8P8/hqdefault.jpg" \* MERGEFORMATINET . The following applet lets you explore Riemann Sums of any function. You can change the bounds and the number of partitions. Follow the examples given on the page, and then use the applet to explore on your own.  HYPERLINK "http://www.calculusapplets.com/riemann.html" Riemann Sums Applet. Note: On this page the author uses Left- and Right- hand sums. These are similar to the sums  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=S%28P%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=T%28P%29" \* MERGEFORMATINET that you have learned, particularly in the case of an increasing (or decreasing) function. Left-hand and Right-hand sums are frequently used in calculations of numerical integrals because it is easy to find the left and right endpoints of each interval, and much more difficult to find the max/min of the function on each interval. The difference is not always important from a numerical approximation standpoint; as you increase the number of partitions, you should see the Left-hand and Right-hand sums converging to the same value. Try this in the applet to see for yourself. Review Questions In problems #17 , use Riemann Sums to approximate the areas under the curves. Consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%202%20-%20x" \* MERGEFORMATINET  from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET  to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202." \* MERGEFORMATINET  Use Riemann Sums with four subintervals of equal lengths. Choose the midpoints of each subinterval as the sample points. Repeat problem #1 using geometry to calculate the exact area of the region under the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%202%20-%20x" \* MERGEFORMATINET from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET  to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202." \* MERGEFORMATINET  (Hint: Sketch a graph of the region and see if you can compute its area using area measurement formulas from geometry.) Repeat problem #1 using the definition of the definite integral to calculate the exact area of the region under the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%202%20-%20x" \* MERGEFORMATINET  from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET  to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%20-%20x" \* MERGEFORMATINET  from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201" \* MERGEFORMATINET  to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%204." \* MERGEFORMATINET  Use Riemann Sums with five subintervals of equal lengths. Choose the left endpoint of each subinterval as the sample points. Repeat problem #4 using the definition of the definite integral to calculate the exact area of the region under the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%20-%20x" \* MERGEFORMATINET  from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201" \* MERGEFORMATINET  to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%204." \* MERGEFORMATINET  Consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%203%20x%5E2." \* MERGEFORMATINET  Compute the Riemann Sum of f on [0, 1] under each of the following situations. In each case, use the right endpoint as the sample points. Two sub-intervals of equal length. Five sub-intervals of equal length. Ten sub-intervals of equal length. Based on your answers above, try to guess the exact area under the graph of f on [0, 1]. Consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20e%5Ex" \* MERGEFORMATINET . Compute the Riemann Sum of f on [0, 1] under each of the following situations. In each case, use the right endpoint as the sample points. Two sub-intervals of equal length. Five sub-intervals of equal length. Ten sub-intervals of equal length. Based on your answers above, try to guess the exact area under the graph of f on [0, 1]. Find the net area under the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20-%20x" \* MERGEFORMATINET ;  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-1" \* MERGEFORMATINET  to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201." \* MERGEFORMATINET  (Hint: Sketch the graph and check for symmetry.) Find the total area bounded by the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3%20-%20x" \* MERGEFORMATINET  and the x-axis, from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D-1" \* MERGEFORMATINET  to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201." \* MERGEFORMATINET  Use your knowledge of geometry to evaluate the following definite integral:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E3%20%5Csqrt%7B9%20-%20x%5E2%7D%20dx" \* MERGEFORMATINET  (Hint: set  EMBED Equation.DSMT4  and square both sides to see if you can recognize the region from geometry.) Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%202" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%202" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%202" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%2010.08" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%2015.5" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%201.875" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%201.32" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%201.15" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%201" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%202.18" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%201.89" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%201.80" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%20e%5E1%20-1%20%5Capprox%201.71" \* MERGEFORMATINET  The graph is symmetric about the origin; hence net Area = 0 Area = 1/2 The graph is that of a quarter circle of radius 3; hence Area =  EMBED Equation.DSMT4  7.5 Evaluating Definite Integrals Learning Objectives Use antiderivatives to evaluate definite integrals Use the Mean Value Theorem for integrals to solve problems Use general rules of integrals to solve problems Introduction In the Lesson on Definite Integrals, we evaluated definite integrals using the limit definition. This process was long and tedious. In this lesson we will learn some practical ways to evaluate definite integrals. We begin with a theorem that provides an easier method for evaluating definite integrals. Newton discovered this method that uses antiderivatives to calculate definite integrals. Theorem: If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is continuous on the closed interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint_a%5Eb%20f%28x%29%20dx%20%3D%20F%28b%29%20-%20F%28a%29%2C" \* MERGEFORMATINET  where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F" \* MERGEFORMATINET is any antiderivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET  We sometimes use the following shorthand notation to indicate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_a%5Eb%20f%28x%29%20dx%20%3D%20F%28b%29%20-%20F%28a%29%3A" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint_a%5Eb%20f%28x%29%20dx%20%3D%20F%28x%29%20%5Cbig%20%5D_a%5Eb." \* MERGEFORMATINET  The proof of this theorem is included at the end of this lesson. Theorem 4.1 is usually stated as a part of the Fundamental Theorem of Calculus, a theorem that we will present in the Lesson on the Fundamental Theorem of Calculus. For now, the result provides a useful and efficient way to compute definite integrals. We need only find an antiderivative of the given function in order to compute its integral over the closed interval. It also gives us a result with which we can now state and prove a version of the Mean Value Theorem for integrals. But first lets look at a couple of examples. Example 1: Compute the following definite integral:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint_0%5E3%20x%5E3%20dx." \* MERGEFORMATINET  Solution: Using the limit definition we found that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E3%20x%5E3%20dx%20%3D%20%5Cfrac%7B81%7D%20%7B4%7D." \* MERGEFORMATINET We now can verify this using the theorem as follows: We first note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%5E4/4" \* MERGEFORMATINET is an antiderivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E3." \* MERGEFORMATINET Hence we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint_0%5E3%20x%5E3%20dx%20%3D%20%5Cfrac%7Bx%5E4%7D%20%7B4%7D%20%5Cbigg%20%5D_0%5E3%20%3D%20%5Cfrac%7B81%7D%20%7B4%7D%20-%20%5Cfrac%7B0%7D%20%7B4%7D%20%3D%20%5Cfrac%7B81%7D%20%7B4%7D." \* MERGEFORMATINET  We conclude the lesson by stating the rules for definite integrals, most of which parallel the rules we stated for the general indefinite integrals.  Given these rules together with Theorem 4.1, we will be able to solve a great variety of definite integrals. Example 2: Compute  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_%7B-2%7D%5E%7B2%7D%20%28x%20-%20%5Csqrt%7Bx%7D%29%20dx." \* MERGEFORMATINET  Solution:  Example 3: Compute  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_%7B0%7D%5E%7B%5Cfrac%7B%5Cpi%7D%20%7B2%7D%7D%20%28x%20%2B%20%5Ccos%20x%29%20dx." \* MERGEFORMATINET  Solution:  Lesson Summary We used antiderivatives to evaluate definite integrals. We used the Mean Value Theorem for integrals to solve problems. We used general rules of integrals to solve problems. Proof of Theorem 4.1 We first need to divide  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D" \* MERGEFORMATINET into  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET sub-intervals of length  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Ctriangle%20x%20%3D%20%5Cfrac%7Bb%20-%20a%7D%20%7Bn%7D" \* MERGEFORMATINET . We let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_0%20%3D%20a%2C%20x_1%2C%20x_2%2C%20%5Cldots%2C%20x_n%20%3D%20b" \* MERGEFORMATINET be the endpoints of these sub-intervals. Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F" \* MERGEFORMATINET be any antiderivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET  Consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28b%29%20-%20F%28a%29%20%3D%20F%28x_n%29%20-%20F%28x_0%29." \* MERGEFORMATINET  We will now employ a method that will express the right side of this equation as a Riemann Sum. In particular,  Note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F" \* MERGEFORMATINET is continuous. Hence, by the Mean Value Theorem, there exist  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c_i%20%5Cin%20%5Bx_i%20-%201%2C%20x_i%5D" \* MERGEFORMATINET  such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x_i%29%20-%20F%28x_%7Bi%20-%201%7D%29%20%3D%20F%27%28c_i%29%20%28x_i%20-%20x_%7Bi%20-%201%7D%29%20%3D%20%20f%28c_i%29%20%5Ctriangle%20x." \* MERGEFORMATINET  Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20F%28b%29%20-%20F%28a%29%20%3D%20%5Csum_1%5En%20F%27%28c_i%29%20%28x_i%20-%20x_%7Bi%20-%201%7D%29%20%3D%20%5Csum_1%5En%20f%28c_i%29%20%5Ctriangle%20x." \* MERGEFORMATINET  Taking the limit of each side as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%5Crightarrow%20%5Cinfty" \* MERGEFORMATINET we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Clim_%7Bn%20%5Cto%20%5C%20%5Cinfty%7D%20%5BF%28b%29%20-%20F%28a%29%5D%20%3D%20%5Clim_%7Bn%20%5Cto%20%5C%20%5Cinfty%7D%20%5Csum_1%5En%20f%28c_i%29%20%5Ctriangle%20x." \* MERGEFORMATINET  We note that the left side is a constant and the right side is our definition for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_a%5Eb%20f%28x%29%20dx" \* MERGEFORMATINET . Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20F%28b%29%20-%20F%28a%29%20%3D%20%5Clim_%7Bn%20%5Cto%20%5C%20%5Cinfty%7D%20%5Csum_1%5En%20f%28c_i%29%20%5Ctriangle%20x%20%3D%20%5Cint_a%5Eb%20f%28x%29%20dx." \* MERGEFORMATINET  Proof of Theorem 4.2 Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x%29%20%3D%20%5Cint_a%5Ex%20f%28x%29%20dx." \* MERGEFORMATINET  By the Mean Value Theorem for derivatives, there exists  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c%20%5Cin%20%5Ba%2C%20b%5D" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20F%27%28c%29%20%3D%20%5Cfrac%7BF%28b%29%20-%20F%28a%29%7D%20%7Bb%20-%20a%7D." \* MERGEFORMATINET  From Theorem 4.1 we have that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F" \* MERGEFORMATINET is an antiderivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f." \* MERGEFORMATINET Hence,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%28x%29%20%3D%20f%28x%29" \* MERGEFORMATINET and in particular,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%28c%29%20%3D%20f%28c%29." \* MERGEFORMATINET Hence, by substitution we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20f%28c%29%20%3D%20%5Cfrac%7BF%28b%29%20-%20F%28a%29%7D%20%7Bb%20-%20a%7D." \* MERGEFORMATINET  Note that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28a%29%20%3D%20%5Cint_a%5Ea%20f%28x%29%20dx%20%3D%200." \* MERGEFORMATINET  Hence we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20f%28c%29%20%3D%20%5Cfrac%7BF%28b%29%20-%200%7D%20%7Bb%20-%20a%7D%20%3D%20%5Cfrac%7BF%28b%29%7D%20%7Bb%20-%20a%7D%2C" \* MERGEFORMATINET  and by our definition of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x%29" \* MERGEFORMATINET  we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20f%28c%29%20%3D%20%5Cfrac%7B1%7D%20%7Bb%20-%20a%7D%20F%28b%29%20%3D%20%20%5Cfrac%7B1%7D%20%7Bb%20-%20a%7D%20%5Cint_a%5Eb%20f%28x%29%20dx." \* MERGEFORMATINET  This theorem allows us to find for positive functions a rectangle that has base  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET and height  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28c%29" \* MERGEFORMATINET such that the area of the rectangle is the same as the area given by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_a%5Eb%20f%28x%29%20dx." \* MERGEFORMATINET In other words,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28c%29" \* MERGEFORMATINET is the average function value over  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D." \* MERGEFORMATINET  Review Questions In problems #18, use antiderivatives to compute the definite integral.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_4%5E9%20%28%5Cfrac%7B3%7D%20%7B%5Csqrt%7Bx%7D%7D%29%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E1%20%28t%20-%20t%5E2%29dt" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_2%5E5%20%28%5Cfrac%7B1%7D%20%7B%5Csqrt%7Bx%7D%7D%20%2B%20%5Cfrac%7B1%7D%20%7B%5Csqrt%7B2%7D%7D%29%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E1%204%28x%5E2%20-%201%29%20%28x%5E2%20%2B%201%29dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_2%5E8%20%28%5Cfrac%7B4%7D%20%7Bx%7D%20%2B%20x%5E2%20%2B%20x%29%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_2%5E4%20%28e%5E%7B3x%7D%29dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_1%5E4%20%5Cfrac%7B2%7D%20%7Bx%20%2B%203%7Ddx" \* MERGEFORMATINET  Find the average value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%5Csqrt%7Bx%7D" \* MERGEFORMATINET  over [1, 9]. If f is continuous and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_1%5E4%20f%28x%29dx%20%3D%209%2C" \* MERGEFORMATINET  show that f takes on the value 3 at least once on the interval [1, 4]. Your friend states that there is no area under the curve of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%5Csin%20x" \* MERGEFORMATINET  on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C%202%5Cpi%5D" \* MERGEFORMATINET  since he computed  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_0%5E%7B2%5Cpi%7D%20%5Csin%20x%20dx%20%3D%200." \* MERGEFORMATINET  Is he correct? Explain your answer. Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_4%5E9%20%28%5Cfrac%7B3%7D%20%7B%5Csqrt%7Bx%7D%7D%29%20dx%20%3D%206" \* MERGEFORMATINET   EMBED Equation.DSMT4   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_2%5E5%20%28%5Cfrac%7B1%7D%20%7B%5Csqrt%7Bx%7D%7D%20%2B%20%5Cfrac%7B1%7D%20%7B%5Csqrt%7B2%7D%7D%29%20dx%20%3D%202%5Csqrt%7B5%7D%20-%202%5Csqrt%7B2%7D%20%2B%20%5Cfrac%7B3%5Csqrt%7B2%7D%7D%20%7B2%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E1%204%28x%5E2%20-%201%29%20%28x%5E2%20%2B%201%29%20dx%20%3D%20-%20%5Cfrac%7B16%7D%20%7B5%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_2%5E8%20%28%5Cfrac%7B4%7D%20%7Bx%7D%20%2B%20x%5E2%20%2B%20x%29%20dx%20%3D%20%5Cfrac%7B9417%7D%20%7B48%7D%20%5Capprox%20196.19" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_2%5E4%20%28e%5E%7B3x%7D%29%20dx%20%3D%20%5Cfrac%7Be%5E%7B12%7D%20-%20e%5E6%7D%20%7B3%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_1%5E4%20%5Cfrac%7B2%7D%20%7Bx%20%2B%203%7D%20dx%20%3D%202%20%5Cln%207%20-%202%20%5Cln%204" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cfrac%7B13%7D%20%7B6%7D" \* MERGEFORMATINET  Apply the Mean Value Theorem for integrals. He is partially correct. The definite integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E%7B2%5Cpi%7D%20%5Csin%20x%20%5C%2Cdx" \* MERGEFORMATINET  computes the net area under the curve. However, the area between the curve and the x-axis is given by:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20A%20%3D%202%20%5Cint_0%5E%7B%5Cpi%7D%20%5Csin%20x%20%5C%2Cdx%20%3D%20-%20%5Ccos%20x%20%5Cbig%20%5D_0%5E%7B%5Cpi%7D%20%3D%202." \* MERGEFORMATINET  7.6 The Fundamental Theorem of Calculus Learning Objectives Use the Fundamental Theorem of Calculus to evaluate definite integrals Introduction In the Lesson on Evaluating Definite Integrals, we evaluated definite integrals using antiderivatives. This process was much more efficient than using the limit definition. In this lesson we will state the Fundamental Theorem of Calculus and continue to work on methods for computing definite integrals. Fundamental Theorem of Calculus: Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET be continuous on the closed interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D." \* MERGEFORMATINET  If function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F" \* MERGEFORMATINET is defined by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x%29%20%3D%20%5Cint_a%5Ex%20f%28t%29%20dx%2C" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET , then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%28x%29%20%3D%20f%28x%29" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%20%2C%20b%5D." \* MERGEFORMATINET  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g" \* MERGEFORMATINET is any antiderivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint_a%5Eb%20f%28t%29%20dt%20%3D%20g%28b%29%20-%20g%28a%29." \* MERGEFORMATINET  We first note that we have already proven part 2 as Theorem 4.1. The proof of part 1 appears at the end of this lesson. Think about this Theorem. Two of the major unsolved problems in science and mathematics turned out to be solved by calculus which was invented in the seventeenth century. These are the ancient problems: Find the areas defined by curves, such as circles or parabolas. Determine an instantaneous rate of change or the slope of a curve at a point. With the discovery of calculus, science and mathematics took huge leaps, and we can trace the advances of the space age directly to this Theorem. Lets continue to develop our strategies for computing definite integrals. We will illustrate how to solve the problem of finding the area bounded by two or more curves. Example 1: Find the area between the curves of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28x%29%20%3D%20x%5E3." \* MERGEFORMATINET  Solution: We first observe that there are no limits of integration explicitly stated here. Hence we need to find the limits by analyzing the graph of the functions.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123789" \* MERGEFORMATINET  We observe that the regions of interest are in the first and third quadrants from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-1" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201." \* MERGEFORMATINET We also observe the symmetry of the graphs about the origin. From this we see that the total area enclosed is  Example 2: Find the area between the curves of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%7Cx%20-%201%7C" \* MERGEFORMATINET and the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x-" \* MERGEFORMATINET axis from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-1" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203." \* MERGEFORMATINET  Solution: We observe from the graph that we will have to divide the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B-1%2C%203%5D" \* MERGEFORMATINET into subintervals  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B-1%2C%201%5D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B1%2C%203%5D." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123776" \* MERGEFORMATINET  Hence the area is given by  Example 3: Find the area enclosed by the curves of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%20%2B%202%20x%20%2B%201" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=g%28x%29%20%3D%20-x%5E2%20-%202%20x%20%2B%201." \* MERGEFORMATINET  Solution: The graph indicates the area we need to focus on.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123792" \* MERGEFORMATINET   Before providing another example, lets look back at the first part of the Fundamental Theorem. If function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F" \* MERGEFORMATINET is defined by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%29%20%3D%20%5Cint_a%5Ex%20f%28t%29dt%2C" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%28x%29%20%3D%20f%28x%29" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%20%2C%20b%5D." \* MERGEFORMATINET Observe that if we differentiate the integral with respect to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%2C" \* MERGEFORMATINET we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cfrac%7Bd%7D%20%7Bdx%7D%20%5Cint_a%5Ex%20f%28t%29dt%20%3D%20F%27%28x%29%20%3D%20f%28x%29." \* MERGEFORMATINET  This fact enables us to compute derivatives of integrals as in the following example. Example 4: Use the Fundamental Theorem to find the derivative of the following function:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20g%28x%29%20%3D%20%5Cint_0%5Ex%281%20%2B%20%5Csqrt%5B3%5D%7Bt%7D%29dt." \* MERGEFORMATINET  Solution: While we could easily integrate the right side and then differentiate, the Fundamental Theorem enables us to find the answer very routinely.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20g%27%28x%29%20%3D%20%5Cfrac%7Bd%7D%20%7Bdx%7D%20%5Cint_0%5Ex%281%20%2B%20%5Csqrt%5B3%5D%7Bt%7D%29dt%20%3D%201%20%2B%20%5Csqrt%5B3%5D%7Bx%7D." \* MERGEFORMATINET  This application of the Fundamental Theorem becomes more important as we encounter functions that may be more difficult to integrate such as the following example. Example 5: Use the Fundamental Theorem to find the derivative of the following function:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20g%28x%29%20%3D%20%5Cint_2%5Ex%28t%5E2%20%5Ccos%20t%29dt." \* MERGEFORMATINET  Solution: In this example, the integral is more difficult to evaluate. The Fundamental Theorem enables us to find the answer routinely.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20g%27%28x%29%20%3D%20%5Cfrac%7Bd%7D%20%7Bdx%7D%20%5Cint_2%5Ex%28t%5E2%20%5Ccos%20t%29dt%20%3D%20x%5E2%20%5Ccos%20x." \* MERGEFORMATINET  Lesson Summary 1. We used the Fundamental Theorem of Calculus to evaluate definite integrals. Fundamental Theorem of Calculus Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET be continuous on the closed interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D." \* MERGEFORMATINET  If function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F" \* MERGEFORMATINET is defined by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x%29%20%3D%20%5Cint_a%5Ex%20f%28t%29dt" \* MERGEFORMATINET , on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%28x%29%20%3D%20f%28x%29%2C" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D." \* MERGEFORMATINET  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g" \* MERGEFORMATINET is any antiderivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint_a%5Ebf%28t%29dt%20%3D%20g%28b%29%20-%20g%28a%29." \* MERGEFORMATINET  We first note that we have already proven part 2 as Theorem 4.1. Proof of Part 1. 1. Consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x%29%20%3D%20%5Cint_a%5Ex%20f%28t%29%20dt%2C" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D." \* MERGEFORMATINET  2.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%2C%20c%20%5Cin%20%5Ba%2C%20b%5D%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c%20%3C%20x." \* MERGEFORMATINET  Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_a%5Ex%20f%28t%29%20dt%20%3D%20%5Cint_a%5Ec%20f%28t%29%20dt%20%2B%20%5Cint_c%5Ex%20f%28t%29%20dt" \* MERGEFORMATINET by our rules for definite integrals. 3. Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_a%5Ex%20f%28t%29%20dt%20-%20%5Cint_a%5Ec%20f%28t%29%20dt%20%3D%20%5Cint_c%5Ex%20f%28t%29%20dt" \* MERGEFORMATINET . Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%28x%29%20-%20F%28c%29%20%3D%20%5Cint_c%5Ex%20f%28t%29%20dt." \* MERGEFORMATINET  4. Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is continuous on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%2C%20c%20%5Cin%20%5Ba%2C%20b%5D%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c%20%3C%20x" \* MERGEFORMATINET then we can select  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%2C%20v%20%5Cin%20%5Bc%2C%20v%5D" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28u%29" \* MERGEFORMATINET is the minimum value of and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28v%29" \* MERGEFORMATINET is the maximum value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Bc%2C%20x%5D." \* MERGEFORMATINET Then we can consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28u%29%20%28x%20-%20c%29" \* MERGEFORMATINET as a lower sum and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28v%29%20%28x%20-%20c%29" \* MERGEFORMATINET as an upper sum of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x." \* MERGEFORMATINET Hence 5.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28u%29%20%28x%20-%20c%29%20%5Cle%20%5Cint_c%5Ex%20f%28t%29%20dt%20%5Cle%20f%28v%29%20%28x%20-%20c%29." \* MERGEFORMATINET  6. By substitution, we have:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=f%28u%29%20%28x%20-%20c%29%20%5Cle%20F%28x%29%20-F%28c%29%20%5Cle%20f%28v%29%20%28x%20-%20c%29." \* MERGEFORMATINET  7. By division, we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20f%28u%29%20%5Cle%20%5Cfrac%7BF%28x%29%20-%20F%28c%29%7D%20%7Bx%20-%20c%7D%20%5Cle%20f%28v%29." \* MERGEFORMATINET  8. When  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET is close to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c%2C" \* MERGEFORMATINET then both  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28u%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28v%29" \* MERGEFORMATINET are close to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28c%29" \* MERGEFORMATINET by the continuity of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET  9. Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Clim_%7Bx%20%5Cto%20c%5E%2B%7D%20%5Cfrac%7BF%28x%29%20-%20F%28c%29%7D%20%7Bx%20-%20c%7D%20%3D%20f%28c%29." \* MERGEFORMATINET Similarly, if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3C%20c%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Clim_%7Bx%20%5Cto%20c%5E-%7D%20%5Cfrac%7BF%28x%29%20-%20F%28c%29%7D%20%7Bx%20-%20c%7D%20%3D%20f%28c%29." \* MERGEFORMATINET Hence,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Clim_%7Bx%20%5Cto%20c%7D%20%5Cfrac%7BF%28x%29%20-%20F%28c%29%7D%20%7Bx%20-%20c%7D%20%3D%20f%28c%29." \* MERGEFORMATINET  10. By the definition of the derivative, we have that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20F%27%28c%29%20%3D%20%5Clim_%7Bx%20%5Cto%20c%7D%20%5Cfrac%7BF%28x%29%20-%20F%28c%29%7D%20%7Bx%20-%20c%7D%20%3D%20f%28c%29" \* MERGEFORMATINET for every  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=c%20%5Cin%20%5Ba%2C%20b%5D." \* MERGEFORMATINET Thus,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F" \* MERGEFORMATINET is an antiderivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D." \* MERGEFORMATINET  Multimedia Link For a video presentation of the Fundamental Theorem of Calculus (15.0), see  HYPERLINK "http://www.youtube.com/watch?v=PGmVvIglZx8" Fundamental Theorem of Calculus, Part 1 (9:26) INCLUDEPICTURE "http://i1.ytimg.com/vi/PGmVvIglZx8/hqdefault.jpg" \* MERGEFORMATINET . Review Questions In problems #14, sketch the graph of the function f(x) in the interval [a, b]. Then use the Fundamental Theorem of Calculus to find the area of the region bounded by the graph and the x-axis. (Hint: Examine the graph of the function and divide the interval accordingly.)  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%202%20x%20%2B%203%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C%204%5D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20e%5Ex%20%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C%202%5D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%20%2B%20x%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B1%2C%203%5D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%20-%20x%2C" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C%202%5D" \* MERGEFORMATINET  In problems #57 use antiderivatives to compute the definite integral. (Hint: Examine the graph of the function and divide the interval accordingly.)  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_%7B-1%7D%5E%7B%2B1%7D%20%7Cx%7C%20dx" \* MERGEFORMATINET   EMBED Equation.DSMT4   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_%7B-2%7D%5E%7B%2B4%7D%20%5Cleft%20%5B%7Cx%20-%201%7C%20%2B%20%7Cx%20%2B%201%7C%20%5Cright%5D%20dx" \* MERGEFORMATINET  In problems #810, find the area between the graphs of the functions.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 , on the interval (0, 3) (Hint: you will need to add 2 integrals) Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%20%3D%2028" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123779" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%20e%5E2%20-%201" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123782" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%20%5Cfrac%7B38%7D%20%7B3%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123784" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5C%3B%5Cmathrm%7BArea%7D%20%3D%201" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123777" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_%7B-1%7D%5E%7B%2B1%7D%20%7Cx%7C%20dx%20%3D%201" \* MERGEFORMATINET  18.75  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_%7B-2%7D%5E%7B%2B4%7D%20%5Cbigg%20%5B%7Cx%20-%201%7C%20%2B%20%7Cx%20%2B%201%7C%20%5Cbigg%20%5D%20dx%20%3D%2022" \* MERGEFORMATINET  Area = 1/6  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123783" \* MERGEFORMATINET  Area = 32/3  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123780" \* MERGEFORMATINET  Area =  EMBED Equation.DSMT4   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123790" \* MERGEFORMATINET  7.7 Integration by Substitution Learning Objectives Integrate composite functions Use change of variables to evaluate definite integrals Use substitution to compute definite integrals Introduction In this lesson we will expand our methods for evaluating definite integrals. We first look at a couple of situations where finding antiderivatives requires special methods. These involve finding antiderivatives of composite functions and finding antiderivatives of products of functions. Antiderivatives of Composites Suppose we needed to compute the following integral:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%203x%5E2%20%5Csqrt%7B1%20%2B%20x%5E3%7D%20dx." \* MERGEFORMATINET  Our rules of integration are of no help here. We note that the integrand is of the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28g%28x%29%29%20%2A%20g%27%28x%29" \* MERGEFORMATINET where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28x%29%20%3D%201%20%2B%20x%5E3" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20%5Csqrt%7Bx%7D." \* MERGEFORMATINET  Since we are looking for an antiderivative  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F" \* MERGEFORMATINET of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%2C" \* MERGEFORMATINET and we know that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%27%20%3D%20f%2C" \* MERGEFORMATINET we can re-write our integral as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20%5Csqrt%7B1%20%2B%20x%5E3%7D%20%5Ccdot%203x%5E2%20dx%20%3D%20%5Cfrac%7B2%7D%20%7B3%7D%20%28%5Csqrt%7B1%20%2B%20x%5E3%7D%29%5E%7B%5Cfrac%7B3%7D%20%7B2%7D%7D%20%2B%20C." \* MERGEFORMATINET  In practice, we use the following substitution scheme to verify that we can integrate in this way: 4. Integrate with respect to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET :  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%5Csqrt%7B1%20%2B%20x%5E3%7D%20%5Ccdot%203x%5E2%20dx%20%3D%20%5Cint%20%5Csqrt%7Bu%7D%20du%2C" \* MERGEFORMATINET where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%201%20%2B%20x%5E3" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20du%20%3D%203%20x%5E2%20dx." \* MERGEFORMATINET  3. Change the original integral in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET to an integral in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET : 2. Differentiate both sides so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%203%20x%5E2%20dx." \* MERGEFORMATINET  1. Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%201%20%2B%20x%5E3." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20%5Csqrt%7Bu%7D%20du%20%3D%20%5Cint%20u%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20du%20%3D%20%5Cfrac%7B2%7D%20%7B3%7D%20u%5E%7B%5Cfrac%7B3%7D%20%7B2%7D%7D%20%2B%20C." \* MERGEFORMATINET  5. Change the answer back to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET : While this method of substitution is a very powerful method for solving a variety of problems, we will find that we sometimes will need to modify the method slightly to address problems, as in the following example. Example 1: Compute the following indefinite integral:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20x%5E2%20e%5E%7Bx%5E3%7D%20dx." \* MERGEFORMATINET  Solution: We note that the derivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%5E3" \* MERGEFORMATINET is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=3%20x%5E2" \* MERGEFORMATINET ; hence, the current problem is not of the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20F%27%28g%28x%29%29%20%5Ccdot%20g%27%28x%29dx." \* MERGEFORMATINET But we notice that the derivative is off only by a constant of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=3" \* MERGEFORMATINET and we know that constants are easy to deal with when differentiating and integrating. Hence Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20x%5E3." \* MERGEFORMATINET  Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%203%20x%5E2%20dx." \* MERGEFORMATINET  Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cfrac%7B1%7D%20%7B3%7D%20du%20%3D%20x%5E2%20dx." \* MERGEFORMATINET and we are ready to change the original integral from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET to an integral in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET and integrate:  Changing back to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET , we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20x%5E2%20e%5E%7Bx%5E3%7D%20dx%20%3D%20%5Cfrac%7B1%7D%20%7B3%7D%20e%5E%7Bx%5E3%7D%20%2B%20C." \* MERGEFORMATINET  We can also use this substitution method to evaluate definite integrals. If we attach limits of integration to our first example, we could have a problem such as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint_1%5E4%20%5Csqrt%7B1%20%2B%20x%5E3%7D%20%5Ccdot%203x%5E2%20dx." \* MERGEFORMATINET  The method still works. However, we have a choice to make once we are ready to use the Fundamental Theorem to evaluate the integral. Recall that we found that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%5Csqrt%7B1%20%2B%20x%5E3%7D%20%5Ccdot%203x%5E2%20dx%20%3D%20%5Cint%20%5Csqrt%7Bu%7D%20du" \* MERGEFORMATINET for the indefinite integral. At this point, we could evaluate the integral by changing the answer back to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET or we could evaluate the integral in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u." \* MERGEFORMATINET But we need to be careful. Since the original limits of integration were in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET , we need to change the limits of integration for the equivalent integral in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u." \* MERGEFORMATINET Hence,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_1%5E4%20%5Csqrt%7B1%20%2B%20x%5E3%7D%20%5Ccdot%203x%5E2%20dx%20%20%3D%20%5Cint_%7Bu%20%3D%202%7D%5E%7B65%7D%20%5Csqrt%7Bu%20du%7D%2C" \* MERGEFORMATINET where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%201%20%2B%20x%5E3" \* MERGEFORMATINET   Integrating Products of Functions We are not able to state a rule for integrating products of functions,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20f%28x%29%20g%28x%29%20dx" \* MERGEFORMATINET but we can get a relationship that is almost as effective. Recall how we differentiated a product of functions:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cfrac%7Bd%7D%20%7Bdx%7D%20f%28x%29%20g%28x%29%20%3D%20f%28x%29%20g%27%28x%29%20%2B%20g%28x%29%20f%27%28x%29." \* MERGEFORMATINET  So by integrating both sides we get  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%5Bf%28x%29%20g%27%28x%29%20%2B%20g%28x%29%20f%27%28x%29%5D%20dx%20%3D%20f%28x%29%20g%28x%29%2C" \* MERGEFORMATINET or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20f%28x%29%20g%27%28x%29%20dx%20%3D%20f%28x%29%20g%28x%29%20-%20%5Cint%20g%28x%29%20f%27%28x%29." \* MERGEFORMATINET  In order to remember the formula, we usually write it as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20u%20dv%20%3D%20uv%20-%20%5Cint%20v%20du." \* MERGEFORMATINET  We refer to this method as integration by parts. The following example illustrates its use. Example 2: Use integration by parts method to compute  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20x%20e%5Ex%20dx." \* MERGEFORMATINET  Solution: We note that our other substitution method is not applicable here. But our integration by parts method will enable us to reduce the integral down to one that we can easily evaluate. Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20x" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv%20%3D%20e%5Ex%20dx" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20dx" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=v%20%3D%20e%5Ex" \* MERGEFORMATINET  By substitution, we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20x%20e%5Ex%20dx%20%3D%20x%20e%5Ex%20-%20%5Cint%20e%5Ex%20dx." \* MERGEFORMATINET  We can easily evaluate the integral and have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint%20x%20e%5Ex%20dx%20%3D%20x%20e%5Ex%20-%20%5Cint%20e%5Ex%20dx%20%3D%20x%20e%5Ex%20-%20e%5Ex%20%2B%20C." \* MERGEFORMATINET  And should we wish to evaluate definite integrals, we need only to apply the Fundamental Theorem to the antiderivative. Lesson Summary We integrated composite functions. We used change of variables to evaluate definite integrals. We used substitution to compute definite integrals. Review Questions Compute the integrals in problems #18.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20x%20%5Cln%20x%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%5Cfrac%20%7Bx%7D%7B%5Csqrt%20%7B2x%2B1%7D%7Ddx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E1%20x%5E3%20%5Csqrt%7B1-x%5E2%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20x%20%5Ccos%20x%20dx%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E1%20x%5E2%20%5Csqrt%7Bx%5E3%2B%209%7D%20dx%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%5Cleft%20%28%20%5Cfrac%7B1%7D%7Bx%5E2%7D%20%5Ccdot%20e%5E%5Cfrac%20%7B1%7D%7Bx%7D%20%5Cright%20%29dx%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20x%5E3%20e%5E%7Bx%5E2%7D%20dx%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_1%5Ee%20%5Cfrac%20%7B1%7D%7Bx%7D%20dx" \* MERGEFORMATINET  Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20x%20%5Cln%20x%20dx%20%3D%20%5Cfrac%7Bx%5E2%20%282%20%5Cln%20x%20-%201%29%7D%20%7B4%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%5Cfrac%20%7Bx%7D%7B%5Csqrt%20%7B2x%2B1%7D%7Ddx%20%3D%20%5Cfrac%20%7B%28x-1%29%5Csqrt%20%7B2x%2B1%7D%7D%7B3%7D%20%2B%20c" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E1%20x%5E3%20%5Csqrt%7B1-x%5E2%7D%20dx%20%3D%20%5Cfrac%20%7B2%7D%20%7B15%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20x%20%5Ccos%20x%20dx%20%3D%20x%20%5Csin%20x%20%2B%20%5Ccos%20x%20%2B%20c%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_0%5E1%20x%5E2%20%5Csqrt%7Bx%5E3%2B9%7D%20dx%20%3D%20%20%5Cfrac%7B2%7D%20%7B9%7D%20%5Cleft%20%5B%7B10%5E%5Cfrac%20%7B3%7D%7B2%7D%7D%20-%20%7B27%7D%20%5Cright%5D%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20%5Cleft%20%28%20%5Cfrac%7B1%7D%7Bx%5E2%7D%20%5Ccdot%20e%5E%5Cfrac%20%7B1%7D%7Bx%7D%20%5Cright%20%29dx%20%3D%20-e%5E%20%5Cfrac%20%7B1%7D%7Bx%7D%20%2B%20c%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint%20x%5E3%20e%5E%7Bx%5E2%7D%20dx%20%3D%20%5Cfrac%20%7B1%7D%7B2%7D%20e%5E%7Bx%5E2%7D%20%28x%5E2-1%29%2B%20c" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_1%5Ee%20%5Cfrac%20%7B1%7D%7Bx%7D%20dx%20%3D%201" \* MERGEFORMATINET  Integration by Substitution Practice Find the indefinite integral. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14. Find the equation of the function  EMBED Equation.DSMT4  whose graph passes through the point  EMBED Equation.DSMT4  and whose derivative is  EMBED Equation.DSMT4 . Answers: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  More Substitution Practice Integrate! 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  5.)  EMBED Equation.DSMT4  6.)  EMBED Equation.DSMT4  7.) Find the equation of the function  EMBED Equation.DSMT4  whose graph passes through the point  EMBED Equation.DSMT4  and whose derivative is  EMBED Equation.DSMT4 . Answers: 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  5.)  EMBED Equation.DSMT4  6.)  EMBED Equation.DSMT4  7.)  EMBED Equation.DSMT4   More Substitution Practice Integrate! Some of these require the general power rule (substitution), others do not. 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  5.)  EMBED Equation.DSMT4  6.)  EMBED Equation.DSMT4  7.)  EMBED Equation.DSMT4  Answers: 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  5.)  EMBED Equation.DSMT4  6.)  EMBED Equation.DSMT4  7.)  EMBED Equation.DSMT4   7.8 Numerical Integration Learning Objectives Use the Trapezoidal Rule to solve problems Estimate errors for the Trapezoidal Rule Use Simpsons Rule to solve problems Estimate Errors for Simpsons Rule Introduction Recall that we used different ways to approximate the value of integrals. These included Riemann Sums using left and right endpoints, as well as midpoints for finding the length of each rectangular tile. In this lesson we will learn two other methods for approximating integrals. The first of these, the Trapezoidal Rule, uses areas of trapezoidal tiles to approximate the integral. The second method, Simpsons Rule, uses parabolas to make the approximation. Trapezoidal Rule Lets recall how we would use the midpoint rule with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%204" \* MERGEFORMATINET rectangles to approximate the area under the graph of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%20%2B%201" \* MERGEFORMATINET from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123775" \* MERGEFORMATINET  If instead of using the midpoint value within each sub-interval to find the length of the corresponding rectangle, we could have instead formed trapezoids by joining the maximum and minimum values of the function within each sub-interval:  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123778" \* MERGEFORMATINET  The area of a trapezoid is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20A%20%3D%20%5Cfrac%7Bh%28b_1%20%2B%20b_2%29%7D%20%7B2%7D" \* MERGEFORMATINET , where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=b_1" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=b_2" \* MERGEFORMATINET are the lengths of the parallel sides and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=h" \* MERGEFORMATINET is the height. In our trapezoids the height is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20x" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=b_1" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=b_2" \* MERGEFORMATINET are the values of the function. Therefore in finding the areas of the trapezoids we actually average the left and right endpoints of each sub-interval. Therefore a typical trapezoid would have the area  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20A%20%3D%20%5Cfrac%7B%5Ctriangle%20x%7D%20%7B2%7D%20%5Cleft%20%28f%28x_%7Bi%20-%201%7D%29%20%2B%20f%28x_i%29%20%5Cright%29." \* MERGEFORMATINET  To approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_a%5Eb%20f%28x%29dx" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET of these trapezoids, we have  Example 1: Use the Trapezoidal Rule to approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B3%7D%20x%5E2%20dx" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%206" \* MERGEFORMATINET . Solution: We find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20x%20%3D%20%5Cfrac%7Bb-a%7D%7Bn%7D%20%3D%20%5Cfrac%7B3-0%7D%7B6%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D." \* MERGEFORMATINET   Of course, this estimate is not nearly as accurate as we would like. For functions such as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29%20%3D%20x%5E2%2C" \* MERGEFORMATINET we can easily find an antiderivative with which we can apply the Fundamental Theorem that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B3%7D%20x%5E2dx%20%3D%20%5Cfrac%7Bx%5E3%7D%7B3%7D%5CBigg%5D_0%5E3%20%3D%209." \* MERGEFORMATINET But it is not always easy to find an antiderivative. Indeed, for many integrals it is impossible to find an antiderivative. Another issue concerns the questions about the accuracy of the approximation. In particular, how large should we take n so that the Trapezoidal Estimate for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B3%7D%20x%5E2%20dx" \* MERGEFORMATINET is accurate to within a given value, say  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0.001" \* MERGEFORMATINET ? As with our Linear Approximations in the Lesson on Approximation Errors, we can state a method that ensures our approximation to be within a specified value. Error Estimates for Simpson's Rule We would like to have confidence in the approximations we make. Hence we can choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET to ensure that the errors are within acceptable boundaries. The following method illustrates how we can choose a sufficiently large  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n." \* MERGEFORMATINET  Suppose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7Cf%27%27%28x%29%7C%20%5Cle%20k" \* MERGEFORMATINET for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=a%20%5Cle%20x%20%5Cle%20b." \* MERGEFORMATINET Then the error estimate is given by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%7CError_%7BTrapezoidal%7D%7C%20%5Cle%20%5Cfrac%7Bk%28b-a%29%5E3%7D%7B12n%5E2%7D." \* MERGEFORMATINET  Example 2: Find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET so that the Trapezoidal Estimate for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B3%7Dx%5E2dx" \* MERGEFORMATINET is accurate to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0.001." \* MERGEFORMATINET  Solution: We need to find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7CError_%7BTrapezoidal%7D%7C%20%5Cle%200.001." \* MERGEFORMATINET We start by noting that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7Cf%27%27%28x%29%7C%20%3D%202" \* MERGEFORMATINET for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0%20%5Cle%20x%20%5Cle%203." \* MERGEFORMATINET Hence we can take  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=K%20%3D%202" \* MERGEFORMATINET to find our error bound.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%7CError_%7BTrapezoidal%7D%7C%20%5Cle%20%5Cfrac%7B2%283-0%29%5E3%7D%7B12n%5E2%7D%20%3D%20%5Cfrac%7B54%7D%7B12n%5E2%7D." \* MERGEFORMATINET  We need to solve the following inequality for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET :  Hence we must take  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%2068" \* MERGEFORMATINET to achieve the desired accuracy. From the last example, we see one of the weaknesses of the Trapezoidal Ruleit is not very accurate for functions where straight line segments (and trapezoid tiles) do not lead to a good estimate of area. It is reasonable to think that other methods of approximating curves might be more applicable for some functions. Simpsons Rule is a method that uses parabolas to approximate the curve. Simpsons Rule: As was true with the Trapezoidal Rule, we divide the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2Cb%5D" \* MERGEFORMATINET into  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET sub-intervals of length  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20x%20%3D%20%5Cfrac%7Bb-a%7D%7Bn%7D." \* MERGEFORMATINET We then construct parabolas through each group of three consecutive points on the graph. The graph below shows this process for the first three such parabolas for the case of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%206" \* MERGEFORMATINET sub-intervals. You can see that every interval except the first and last contains two estimates, one too high and one too low, so the resulting estimate will be more accurate.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123786" \* MERGEFORMATINET  Using parabolas in this way produces the following estimate of the area from Simpsons Rule:  We note that it has a similar appearance to the Trapezoidal Rule. However, there is one distinction we need to note. The process of using three consecutive  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_i" \* MERGEFORMATINET to approximate parabolas will require that we assume that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET must always be an even number. Error Estimates for the Trapezoidal Rule As with the Trapezoidal Rule, we have a formula that suggests how we can choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET to ensure that the errors are within acceptable boundaries. The following method illustrates how we can choose a sufficiently large  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n." \* MERGEFORMATINET  Suppose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7Cf%5E4%28x%29%7C%20%5Cle%20k" \* MERGEFORMATINET for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=a%20%5Cle%20x%20%5Cle%20b." \* MERGEFORMATINET Then the error estimate is given by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%7CError_%7Bsimpson%7D%7C%20%5Cle%20%5Cfrac%7Bk%28b-a%29%5E5%7D%7B180n%5E4%7D." \* MERGEFORMATINET  Example 3: a. Use Simpsons Rule to approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B4%7D%5Cfrac%7B1%7D%7Bx%7Ddx" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%206" \* MERGEFORMATINET . Solution: We find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20x%20%3D%20%5Cfrac%7Bb-a%7D%7Bn%7D%20%3D%20%5Cfrac%7B4-1%7D%7B6%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D." \* MERGEFORMATINET   This turns out to be a pretty good estimate, since we know that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint_%7B1%7D%5E%7B4%7D%5Cfrac%7B1%7D%7Bx%7Ddx%20%3D%20%5Cln%20x%20%5Cbigg%20%5D_1%5E4%20%3D%20%5Cln%284%29%20-%20%5Cln%281%29%20%3D%201.3863." \* MERGEFORMATINET  Therefore the error is less than  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0.01" \* MERGEFORMATINET . b. Find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET so that the Simpson Rule Estimate for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B4%7D%20%5Cfrac%7B1%7D%7Bx%7Ddx" \* MERGEFORMATINET is accurate to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0.001." \* MERGEFORMATINET  Solution: We need to find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET such that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7CError_%7Bsimpson%7D%7C%20%5Cle%200.001." \* MERGEFORMATINET We start by noting that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%7Cf%5E4%28x%29%5Cright%7C%20%3D%20%5Cleft%7C%5Cfrac%7B24%7D%7Bx%5E5%7D%5Cright%7C" \* MERGEFORMATINET for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=1%20%5Cle%20x%20%5Cle%204." \* MERGEFORMATINET Hence we can take  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=K%20%3D%2024" \* MERGEFORMATINET to find our error bound:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%7CError_%7Bsimpson%7D%7C%20%5Cle%20%5Cfrac%7B24%284-1%29%5E5%7D%7B180n%5E4%7D%20%3D%20%5Cfrac%7B5832%7D%7B180n%5E4%7D." \* MERGEFORMATINET  Hence we need to solve the following inequality for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET :  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cfrac%7B5832%7D%7B180n%5E4%7D%3C0.001." \* MERGEFORMATINET  We find that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20n%5E4%20%26%20%3E%20%5Cfrac%7B5832%7D%7B180%280.001%29%7D%2C%20%5C%5C%0An%20%26%20%3E%204%20%5Csqrt%20%7B%5Cfrac%20%7B5832%7D%7B180%280.001%29%7D%7D%20%5Capprox%2013.42." \* MERGEFORMATINET  Hence we must take  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%2014" \* MERGEFORMATINET to achieve the desired accuracy. Technology Note: Estimating a Definite Integral with a TI-83/84 Calculator We will estimate the value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cint_1%5E4%20%5Cfrac%7B1%7D%20%7Bx%7Ddx" \* MERGEFORMATINET . Graph the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20f%28x%29%20%3D%20%5Cfrac%7B1%7D%20%7Bx%7D" \* MERGEFORMATINET with the [WINDOW] setting shown below. The graph is shown in the second screen. Press 2nd [CALC] and choose option 7 (see menu below) When the fourth screen appears, press [1] [ENTER] then [4] [ENTER] to enter the lower and upper limits. The final screen gives the estimate, which is accurate to 7 decimal places.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123785" \* MERGEFORMATINET  Lesson Summary We used the Trapezoidal Rule to solve problems. We estimated errors for the Trapezoidal Rule. We used Simpsons Rule to solve problems. We estimated Errors for Simpsons Rule. Multimedia Links For video presentations of Simpson's Rule (21.0), see  HYPERLINK "http://www.youtube.com/watch?v=ns3k-Lz7qWU" Simpson's Rule, Approximate Integration (7:21) INCLUDEPICTURE "http://i3.ytimg.com/vi/ns3k-Lz7qWU/hqdefault.jpg" \* MERGEFORMATINET  and  HYPERLINK "http://www.youtube.com/watch?v=z_AdoS-ab2w" Math Video Tutorials by James Sousa, Simpson's Rule of Numerical Integration (8:48) INCLUDEPICTURE "http://i3.ytimg.com/vi/z_AdoS-ab2w/hqdefault.jpg" \* MERGEFORMATINET . For a video presentation of Newton's Method (21.0), see  HYPERLINK "http://www.youtube.com/watch?v=1uN8cBGVpfs" Newton's Method (7:29) INCLUDEPICTURE "http://i2.ytimg.com/vi/1uN8cBGVpfs/hqdefault.jpg" \* MERGEFORMATINET . Review Question Use the Trapezoidal Rule to approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7Dx%5E2%20e%5E%7B-x%7D%20dx" \* MERGEFORMATINET  with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%208." \* MERGEFORMATINET  Use the Trapezoidal Rule to approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B4%7D%7B%20%5Cln%7D%5Csqrt%20%7Bx%7D%20dx%20" \* MERGEFORMATINET  with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%206." \* MERGEFORMATINET  Use the Trapezoidal Rule to approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7D%5Csqrt%20%7B1%2Bx%5E4%7D%20dx%20" \* MERGEFORMATINET  with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%204." \* MERGEFORMATINET  Use the Trapezoidal Rule to approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B3%7D%5Cfrac%20%7B1%7D%7Bx%7D%20dx%20" \* MERGEFORMATINET  with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%208." \* MERGEFORMATINET  How large should you take n so that the Trapezoidal Estimate for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B3%7D%5Cfrac%20%7B1%7D%7Bx%7D%20dx%20" \* MERGEFORMATINET  is accurate to within .001? Use Simpsons Rule to approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7Dx%5E2%20e%5E%7B-x%7D%20dx" \* MERGEFORMATINET  with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%208." \* MERGEFORMATINET  Use Simpsons Rule to approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B4%7D%5Csqrt%20%7Bx%7D%5C%20%7B%5Cln%7D%5C%20x%20dx%20" \* MERGEFORMATINET  with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%206." \* MERGEFORMATINET  Use Simpsons Rule to approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B2%7D%5Cfrac%20%7B1%7D%7B%5Csqrt%20%7Bx%5E4%2B1%7D%7D%20dx%20" \* MERGEFORMATINET  with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%206." \* MERGEFORMATINET  Use Simpsons Rule to approximate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7D%20%5Csqrt%20%7B1%2Bx%5E4%7D%20dx%20" \* MERGEFORMATINET  with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%204." \* MERGEFORMATINET  How large should you take n so that the Simpson Estimate for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B2%7D%20%7Be%7D%20dx%20" \* MERGEFORMATINET  is accurate to within .00001? Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7Dx%5E2%20e%5E%7B-x%7D%20dx%20%5Capprox%200.16" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B4%7D%7B%20%5Cln%7D%5Csqrt%20%7Bx%7D%20dx%20%5Capprox%201.26" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7D%5Csqrt%20%7B1%2Bx%5E4%7D%20dx%20%5Capprox%201.10" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B3%7D%5Cfrac%20%7B1%7D%7Bx%7D%20dx%20%5Capprox%201.10" \* MERGEFORMATINET  Take  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%2019" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7Dx%5E2%20e%5E%7B-x%7D%20dx%20%5Capprox%200.16" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B4%7D%5Csqrt%20%7Bx%7D%20%7B%20%5Cln%7D%5C%20x%20dx%20%5Capprox%204.28%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B2%7D%5Cfrac%20%7B1%7D%7B%5Csqrt%20%7Bx%5E4%2B1%7D%7D%20dx%20%5Capprox%201.36" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7D%20%5Csqrt%20%7B1%2Bx%5E4%7D%20dx%20%5Capprox%201.09%20" \* MERGEFORMATINET  Take  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%209" \* MERGEFORMATINET  Day 1 Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Integrate. 1) EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4  3) EMBED Equation.DSMT4  4)  EMBED Equation.DSMT4  5)  EMBED Equation.DSMT4  6)  EMBED Equation.DSMT4  7)  EMBED Equation.DSMT4  8)  EMBED Equation.DSMT4  9)  EMBED Equation.DSMT4  10)  EMBED Equation.DSMT4  11) EMBED Equation.DSMT4  12)  EMBED Equation.DSMT4   13)  EMBED Equation.DSMT4  14)  EMBED Equation.DSMT4  15) EMBED Equation.DSMT4  16)  EMBED Equation.DSMT4  17)  EMBED Equation.DSMT4  Day 2 Integrate. 1)  EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4  3)  EMBED Equation.DSMT4  4)  EMBED Equation.DSMT4  5)  EMBED Equation.DSMT4  6)  EMBED Equation.DSMT4  7)  EMBED Equation.DSMT4  8)  EMBED Equation.DSMT4  9)  EMBED Equation.DSMT4  10)  EMBED Equation.DSMT4  11)  EMBED Equation.DSMT4  12)  EMBED Equation.DSMT4  Day 3 Find the function f(x) that has the given derivative and the given initial condition. 1)  EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4  3)  EMBED Equation.DSMT4  4)  EMBED Equation.DSMT4  5)  EMBED Equation.DSMT4  Find the function f(x) that has the given second derivative and satisfies the given conditions. 6)  EMBED Equation.DSMT4  7)  EMBED Equation.DSMT4  Day 4 Integrate the following using U-substitution. 1)  EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4  3)  EMBED Equation.DSMT4  4)  EMBED Equation.DSMT4  5)  EMBED Equation.DSMT4  6)  EMBED Equation.DSMT4   7)  EMBED Equation.DSMT4  8)  EMBED Equation.DSMT4  9)  EMBED Equation.DSMT4  10)  EMBED Equation.DSMT4  Day 5 Integrate the following using U-substitution. 1)  EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4  3)  EMBED Equation.DSMT4  4)  EMBED Equation.DSMT4  5)  EMBED Equation.DSMT4  6)  EMBED Equation.DSMT4   7)  EMBED Equation.DSMT4  8)  EMBED Equation.DSMT4  9) Find the equation of the function f whose graph passes through the point (0,  EMBED Equation.DSMT4 ) and whose derivative is  EMBED Equation.DSMT4  Day 8 Integrate. 1)  EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4  3)  EMBED Equation.DSMT4  4)  EMBED Equation.DSMT4  5)  EMBED Equation.DSMT4  6)  EMBED Equation.DSMT4  7)  EMBED Equation.DSMT4  8)  EMBED Equation.DSMT4  Day 9 Calculus X Name_______________________________ Integrating Exponential Functions 10PTS 7 completion & 3 randomly chosen for accuracy Evaluate. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4   5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4   9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  Day 10 Integrate. 1)  EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4  3)  EMBED Equation.DSMT4  4)  EMBED Equation.DSMT4  5)  EMBED Equation.DSMT4  6)  EMBED Equation.DSMT4  7)  EMBED Equation.DSMT4  8)  EMBED Equation.DSMT4  Day 11 Calculus X Name_______________________________ Integrating Rational Functions 10PTS 7 completion & 3 randomly chosen for accuracy Evaluate. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4   4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4   7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  Day 13 Consider the function  EMBED Equation.DSMT4 . Find a left end approximation for the area between the curve and the x-axis on the interval [0, 2] with n = 4. Find a right end approx. for the area between the curve and the x-axis on the interval [0, 2] with n = 4. Find a midpoint approx. for the area between the curve and the x-axis on the interval [0, 2] with n = 4. Find a left end approximation for the area between the curve and the x-axis on the interval [0, 2] with n = 8. Find a right end approx. for the area between the curve and the x-axis on the interval [0, 2] with n = 8.  Find a midpoint approx. for the area between the curve and the x-axis on the interval [0, 2] with n = 8. What do you think would be the actual area between the curve and the x-axis on the interval [0, 2]? Explain your reasoning. Find a left end approx. for the area between the curve and the line y = 3 on the interval [0, 2] with n = 4. Find a right end approx. for the area between the curve and the line y = 3 on the interval [0, 2] with n = 4. Find a midpoint approx. for the area between the curve and the line y = 3 on the interval [0, 2] with n = 4. Day 14 Calculus X Name_______________________________ More Reimann Sums 10PTS 7 completion & 3 randomly chosen for accuracy Consider the function  EMBED Equation.DSMT4 . 1) Find a left end approximation for the area between the curve and the x-axis on the interval [0, 2] with n = 4. 2) Find a right end approximation for the area between the curve and the x-axis on the interval [0, 2] with n = 4. 3) Find a midpoint approximation for the area between the curve and the x-axis on the interval [0, 2] with n = 4. 4) Find a left end approximation for the area between the curve and the x-axis on the interval [0, 2] with n = 8. 5) Find a right end approximation for the area between the curve and the x-axis on the interval [0, 2] with n = 8.  6) Find a midpoint approximation for the area between the curve and the x-axis on the interval [0, 2] with n = 8. 7) What do you think would be the actual area between the curve and the x-axis on the interval [0, 2]? Explain your reasoning. 8) Find a left end approx. for the area between the curve and the line y = 3 on the interval [0, 2] with n = 4. 9) Find a right end approx. for the area between the curve and the line y = 3 on the interval [0, 2] with n = 4. 10) Find a midpoint approx. for the area between the curve and the line y = 3 on the interval [0, 2] with n = 4. Day 15 Evaluate the definite integral. 1) EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4  3)  EMBED Equation.DSMT4  4)  EMBED Equation.DSMT4  5)  EMBED Equation.DSMT4  6)  EMBED Equation.DSMT4  7)  EMBED Equation.DSMT4  8)  EMBED Equation.DSMT4  9)  EMBED Equation.DSMT4  10)  EMBED Equation.DSMT4  Day 16 Evaluate the definite integral. 1) EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4  3)  EMBED Equation.DSMT4  4)  EMBED Equation.DSMT4  5)  EMBED Equation.DSMT4  6)  EMBED Equation.DSMT4  7)  EMBED Equation.DSMT4  8)  EMBED Equation.DSMT4  9)  EMBED Equation.DSMT4  10)  EMBED Equation.DSMT4  Day 17 Calculus X Name_______________________________ More Definite Integrals Practice As per the new usual, I will be collecting this. It is worth 10 points, 7 for completion and 3 for the correctness of three randomly chosen problems. Evaluate each of the definite integrals. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4   4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4   8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4   11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  Day 18 Calculus X Name_______________________________ Still More Definite Integrals Practice As per the new usual, I will be collecting this. It is worth 10 points, 7 for completion and 3 for the correctness of three randomly chosen problems. Evaluate each of the definite integrals. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4   5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4   9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  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