ࡱ> hj gdo@ 8bjbj p p oo0. 6$j$$$&[&[&[8^[4\$:] LgLgLgLgm\ olwo8!!!!!!$2R|E$&ommooE LgLgZssso HLg$LgsosLsIur~h$Lg.] u`y's&[Upz/sp0=Vr$$ $6oosoooooEE$$1V$s.$$VIndefinite Integrals Chapter intro text here 8.1 Antiderivatives In the previous chapters we learned how to find marginal cost given a cost function, marginal revenue given a revenue function, and marginal profit given a profit function. Is it possible to do the reverse process? Can we find a profit function EMBED Equation.DSMT4 if we are given the marginal profit function,  EMBED Equation.DSMT4 ? Lets take a simple function like  EMBED Equation.DSMT4 . We know from experience that  EMBED Equation.DSMT4  is the derivative of the function  EMBED Equation.DSMT4 , therefore we call  EMBED Equation.DSMT4  the antiderivative of  EMBED Equation.DSMT4  Example 8.1 Given  EMBED Equation.DSMT4 , show that its antiderivative is  EMBED Equation.DSMT4  Solution ( To show  EMBED Equation.DSMT4  is the antiderivative of  EMBED Equation.DSMT4  we need to take the derivative of  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 , Since  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 is the antiderivative of  EMBED Equation.DSMT4 . ( Recall that the derivative of a function tells us about the slope of a line tangent to the graph of the function at some value x = a. The graphs of a few functions of the form  EMBED Equation.DSMT4  are shown in figure 8.1 below. ( Figure 8.1 Graphs of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4    Since the shape of each curve is the same and the curves are just translations of one another, we would predict that the slope of the tangent lines at all values of x = a must be the same for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Figure 8.2 below has the tangent line drawn for both curves at x = 1. ( Figure 8.2 Graphs of tangent lines to  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  at x = 1.  Since the two tangent lines at x = 1 are parallel,  EMBED Equation.DSMT4 . Thus, it seems safe to conclude that  EMBED Equation.DSMT4  for all values of x = a. We can also show this algebraically and see that  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  . So, which function,  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  is the antiderivative of  EMBED Equation.DSMT4 ? The answer is that both  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are antiderivatives of  EMBED Equation.DSMT4 . As a matter of fact, any function of the form EMBED Equation.DSMT4 , where C is some constant, is an antiderivative of  EMBED Equation.DSMT4 . Thus, the most general antiderivative of  EMBED Equation.DSMT4  is EMBED Equation.DSMT4 . To actually find an antiderivative we need to use the following rule. Power Rule for AntiderivativesIf  EMBED Equation.DSMT4  then the most general antiderivative of  EMBED Equation.DSMT4 is  EMBED Equation.DSMT4 , where C is an arbitrary constant. Example 8.2 Find the most general antiderivative of  EMBED Equation.DSMT4 . Solution ( Using the power rule for antiderivatives we obtain  EMBED Equation.DSMT4 . ( This process of antidifferentiation is also know as integration. Integration uses the symbol  EMBED Equation.DSMT4  (called the integral sign) to denote that the antiderivative of a function is to be taken. When the integral sign is used we always find the most general antiderivative of the function that immediately follows the integral sign. The arbitrary constant C that comes from finding the most general antiderivative is called the constant of integration. Example 8.3 Find  EMBED Equation.DSMT4  Solution ( We need to find the most general antiderivative of  EMBED Equation.DSMT4  using the power rule.  EMBED Equation.DSMT4  Notice that the integral sign and the dx are no longer written after the power rule is applied. This is because the operation of integration has been performed. You can think of this in the same manner you do the plus sign when you figure 2 + 2 = 4. What happened to the plus sign on the right side of the equation? It was no longer written because you performed the operation of addition. ( The following rules will help us find antiderivatives of more complicated functions. Let k and C be constants, then1)  EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4 3)  EMBED Equation.DSMT4  Example 8.4 Find the indefinite integral. a)  EMBED Equation.DSMT4  b) EMBED Equation.DSMT4  Solution ( a) Using the rules from above and the power rule we get  EMBED Equation.DSMT4  Differentiating the answer will demonstrate that the answer checks. b)  EMBED Equation.DSMT4  ( The last two antiderivatives we need to investigate at are those that involve the natural logarithm and the exponential function with base e. Recall that  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . This leads us to the following rules. 1)  EMBED Equation.DSMT4  2)  EMBED Equation.DSMT4  Example 8.5 Find the indefinite integral.  EMBED Equation.DSMT4  Solution (  EMBED Equation.DSMT4  EMBED Equation.DSMT4  ( Example 8.6 Find f(x) given EMBED Equation.DSMT4  if  EMBED Equation.DSMT4 . Solution ( First find  EMBED Equation.DSMT4  Since  EMBED Equation.DSMT4  , we get  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4 . ( 8.2 Integration using substitution For differentiation we had special rules like chain rule, product rule, and quotient rule. Since integration is the inverse process of differentiation, it seems reasonable to conclude that special rules exist for integration. One special rule we will study for integration is called substitution, commonly called u-substitution. Integration by substitution allows us to recover functions that required the chain rule to take its derivative. For example, to find the derivative of  EMBED Equation.DSMT4 , we use the chain rule and set  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4 . Thus, when given  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Thus,  EMBED Equation.DSMT4 or equivocally EMBED Equation.3 . But, how can we recover  EMBED Equation.DSMT4 given EMBED Equation.DSMT4 ? We must use substitution. We begin by letting  EMBED Equation.3  and then take the derivative of both sides with respect to x  EMBED Equation.3  and solve the above equation for dx  EMBED Equation.DSMT4 .  Now substitute u for  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  for dx then simplify.  EMBED Equation.3  We can use the general power rule for antiderivatives on the last integral  EMBED Equation.3  Finally we must substitute  EMBED Equation.3  for u and we have recovered  EMBED Equation.DSMT4  from above.  EMBED Equation.3 . It can, however, be difficult to determine what expression to make u. The table below shows different types of problems that involve substitution. In each case let u = f(x). i) EMBED Equation.3 ii)  EMBED Equation.3 iii)  EMBED Equation.3 iv)  EMBED Equation.3  Example 8.7 Find a)  EMBED Equation.3  b)  EMBED Equation.3  c)  EMBED Equation.3  Solution ( a) This is a type iii in the table above. Since  EMBED Equation.DSMT4  we let  EMBED Equation.3 . Now differentiate this equation with respect to x and then solve for dx.  EMBED Equation.3  Now substitute u for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  for dx.  EMBED Equation.3  After simplifying the integral will be completely in terms of u.  EMBED Equation.DSMT4  Now we must substitute  EMBED Equation.DSMT4  for u to obtain the final answer.  EMBED Equation.DSMT4  b) This is a type ii in the table above. Since  EMBED Equation.DSMT4 we let  EMBED Equation.3  and then differentiate both sides of the equation with respect to x  EMBED Equation.3 . Now substitute u for EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  for dx, and the new limits of integration.  EMBED Equation.3  c) This is a type iv in the table above, so let  EMBED Equation.3 , then  EMBED Equation.3  Making all substitutions we get the following integral.  EMBED Equation.3  (  8.3 Applications of Antiderivatives Example 8.8 The rate of change in sales of bicycles at Teds bicycle shop for the year 2001 is given by  EMBED Equation.DSMT4 , where x represents the month number in 2001 (i.e. x = 1 is January, x = 2 is February, x = 12 is December). If the shop sold 28 bicycles in the month of February, find the shops total sales function,  EMBED Equation.DSMT4 . Solution ( Since we are given the rate of change of sales, we can find the total sales function,  EMBED Equation.DSMT4 , by finding EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  We also know that  EMBED Equation.DSMT4  because the shop sold a total of 28 bicycles in the month of February. Thus,  EMBED Equation.DSMT4  So the sales function is EMBED Equation.DSMT4 . ( Example 8.9 A companys Profit function, in dollars, is given by  EMBED Equation.DSMT4  where x is the number of items sold. In addition, the company has a marginal revenue function of  EMBED Equation.DSMT4 . Determine the total cost for the company to produce 10 items. Solution ( Recall that Profit = Revenue Cost, so  EMBED Equation.DSMT4 . Solving this equation for  EMBED Equation.DSMT4  we get  EMBED Equation.DSMT4 . Thus we need a revenue function and profit function to obtain the cost function. We are given  EMBED Equation.DSMT4  and we can find the revenue function by integrating EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Since no revenue is generated when zero items are sold,  EMBED Equation.DSMT4 . Thus  EMBED Equation.DSMT4  So  EMBED Equation.DSMT4 . Now we can find the cost function  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  To find the cost to produce 10 items we need to find  EMBED Equation.DSMT4 . Thus, it costs $838 to produce 10 of the companys items. Sample Quiz Question 8.1 Show that  EMBED Equation.DSMT4  is the antiderivative of  EMBED Equation.DSMT4 . Question 8.2 Find the most general antiderivative of  EMBED Equation.DSMT4 . Question 8.3 Find  EMBED Equation.DSMT4  Question 8.4 Find  EMBED Equation.DSMT4  Question 8.5 Find  EMBED Equation.DSMT4  Question 8.6 Find  EMBED Equation.DSMT4  Question 8.7 Find  EMBED Equation.DSMT4  Question 8.8 Sonbyrne Sunglass Company found that during the month of March its rate of change of sales could be modeled by  EMBED Equation.DSMT4 , where x represents the day of March. If the company knows that 17 pairs of sunglasses were sold on March 30, find the total number of sunglasses the company sold on March 20. Question 8.9 Larry's custom made furniture shop sells twin size bed frames for $250 each. If the shop's marginal cost is represented by  EMBED Equation.DSMT4 and he knows that it costs $1476.50 to produce 1 bed, find the shop's profit, to the nearest dollar, for producing 20 beds. (Assume the revenue function is linear.) Question 8.10 Welch Construction has found that the value of the company tractor is decreasing at a rate given by  EMBED Equation.DSMT4 , where t represents the number of years after purchase. If the company purchased the tractor for $17650, what is the value of the tractor 3 years after purchase? I dont think that this is the way the majority of instructors teach substitution. Consequently, this needs to be rewritten. Redo the solutions to these examples. We need to add in problems that involve solving problems like \int x sqrt(4x+1) dx. Note to authors: This is saved as chpt8_fig1.eps also. 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