ࡱ> BDAs lbjbjVV 4p<<p(||LB4ZAAAAAAACcFtAAGBgggAgAgg=@@!IV>"A]B0B>FdFD@@FA|"g&AAdBF| : AP CALCULUS (BC) NAME _______________________ Chapter 5 The Definite Integral Date _________________________ The first part of this course has been driven by the derivative. The derivative represents an instantaneous rate of change of a quantity with respect to an independent variable. Historically the problem situation tied to the derivative has been motion of a particle, where the velocity is the derivative of position. This branch is called Differential Calculus. The second half of calculus (and this course) investigates how those instantaneous rates of change in a quantity over an interval (usually time) produce an accumulation of some quantity. This branch is called Integral Calculus. Section 5.1 Estimating with Finite Sums 1) Consider an Old Problem: A car travels 150 miles in 3 hours. Set up an solve an equation for the average speed of the car? Sketch a quick graph of distance v. time (include labeled/titled axes). What is the connection between our answer and the graph? What is the connection between our answer and calculus? 2) Now let us do what Leibnitz and Newton must have done; that is to consider the same problem from the other perspective. This means that given the speed of the car, how far will it travel during a certain time interval? At first, we will assume that the car travels at a constant speed, then use what we have learned to extend the idea to varying rates. A car travels at constant rate of 90 kph from 2:00pm to 5:00pm. i) Set up and solve an equation to determine how far will the car travel? ii) Sketch a quick graph of rate v. time (include labeled/titled axes). iii) Is there some connection between our answer and the graph? What happens if the rate varies over the interval? The velocity of a particle moving along the x-axis is given by: v(t) = 4t + 3, t e" 0; where units are: cm/sec and sec. If the particle starts at the origin (x = 0), where is it at t = 5 seconds? If the particle starts at x = -3, where is it at t = 5 seconds? Homework: page 270 # 1, 2 CONTINUE ( c) We next consider the motion of a particle whose velocity is given by a non-linear function. Example 1 (page 264)  Determining Distance Traveled When Velocity Varies A particle starts at x = 0 and moves along the x-axis with velocity v(t) = t2 for time t e" 0. Where is the particle at t = 3? Draw a graph of the function (velocity vs. time) over the given time interval; Partition the time interval [0, 3] into subintervals of length "t; Draw vertical segments at each endpoint of each subinterval; Discussion Choose an arbitrary point on each subinterval; Calculate the function value at each of these points; Calculate area approximations for each subinterval; and sum; ****************************************************************************** For purposes of this lab, we will partition the interval [0, 3] into 6 subintervals of equal length "t = _____ Subintervals: [ ] [ ] [ ] [ ] [ ] [ ] Arbitrary point: Function value: (height) Area: Approximate area = __________. ****************************************************************************** Repeat these calculations using different arbitrary points. Use the reverse side of this activity sheet. Subintervals: [ ] [ ] [ ] [ ] [ ] [ ] Arbitrary point: Function value: (height) Area: Approximate area = __________. ****************************************************************************** Subintervals: [ ] [ ] [ ] [ ] [ ] [ ] Arbitrary point: Function value: (height) Area: Approximate area = __________. ****************************************************************************** Homework: Page 270 # 5, 6, 7, 10, 12 Once you have mastered this technique for approximating area under a curve and hence, distance traveled, you will want to take advantage of your graphing calculators ability to perform some of the less intellectually demanding tasks. Below are the steps to follow: Make certain your rate function is in Y1; Call up one of the appropriate PROGRAMS (AREA1, INTAREA, or RAM); Input the required field (parameters) values; Record all required values. NOTE: You are still (and will always be) responsible for being able to perform a RAM by hand. Therefore you will want to maintain an excellent grasp of the entire procedure and concepts for partitioning an interval for the purpose of approximating the accumulative effect of a rate function over an interval. Section 5.2 Definite Integrals 1) Riemann Sums a) Review Finite Sums as needed b) Creating a Riemann Sum Consider a continuous function, f(x), defined on [a, b]for purposes of this initial discussion, it will be easiest for f(x)  EMBED Equation.DSMT4  0  EMBED Equation.DSMT4  x  EMBED Equation.DSMT4  [a, b]. i. Partition [a, b] into n sub-intervals by choosing P = {x0, x1, x2, xn} ii. In each sub-interval, (xk-1, xk) choose an arbitrary ck iii. For each sub-interval EMBED Equation.DSMT4  is the width of a rectangle; and f(ck) is the height of that rectangle iv. Ak = f(ck)* EMBED Equation.DSMT4  is the area of each rectangle v.  EMBED Equation.DSMT4  is a Riemann Sum for f(x) on [a, b]. c) Connect the general Riemann Sum with the RAMs from pervious section d) Taking the Limit of a Riemann Sum 2) Definition The Definite Integral as a Limit of a Riemann Sum (see page 276) 3) Theorem 1 The Existence of Definite Integrals 4) The Definite Integral using a Regular Partition 5) Notation and Terminology 6) Using the Notation Homework 5.2a: Page 280 # 1, 3, 6 7) Definite Integral and the Area Connection 8) Determine:  EMBED Equation.DSMT4  Homework 5.2b: Page 280 # 7, 13, 16, 17, 23, 26 9) When f(x) is not non-negative 10) Exploration 1 page 279 11) The Integral of a Constant (w/ Proof) 12) Evaluating Definite Integrals and the Graphing Calculator Homework 5.2c: Page 280 # 31, 32, 34, 35, 36, 47 51, 54, 55 Section 5.3 Definite Integrals and Antiderivatives Recall that a definite integral is defined as the limit of a Riemann sum.  EMBED Equation.DSMT4  on [a, b]. 1) How would the value of the definite integral be effected if the limits of integration were reversed?  EMBED Equation.DSMT4 ? {Hint: Think in terms of the definition} 2) Properties of the Definite Integral Order of Integration (above) [a, a] Constant Multiple Sum/Difference Additivity [a, c], [c, b] Max/Min Domination Homework 5.3.1: Page 290 # 1, 3, 5, 37, 48 3) Average Value of a Function How does one determine the average of a collection of data? Now how can one use that to determine the average value of a function? Definition Average (Mean) Value If f is integrable on [a, b], then its average (mean) value on [a, b] is  EMBED Equation.DSMT4  4) Another Mean Value Theorem? The Mean Value Theorem for Definite Integrals If f is continuous on [a, b] , then at some point c  [a, b]  EMBED Equation.DSMT4  Homework 5.3.2: Page 290 # 11, 16, 17 Let us pause for a moment (ok, maybe longer) to consider what we have so far accomplished: We made a connection between area  under a velocity curve and displacement (change in position); We extended that notion to include any accumulation of change based on any rate function; We used rectangles to approximate area under a curve (Riemann Sums); We used the infinite process of limits (of Riemann Sums) to get exact values for area under a curve; We defined the definite integral to be the limit of a Riemann Sum; We then restated the connection between the definite integral and area. Now let us consider what methods are currently available to us for evaluating a definite integral. 5a) Connecting Differential and Integral Calculus (see Handout) Read page 288; Exploration 2 (page 289) 5b) Evaluating an Integral Using Antiderivatives Homework 5.3.3: Page 290 # 19, 21, 22, 23, 27, 29,30, 35 Section 5.4 Fundamental Theorem of Calculus 1) Fundamental Theorem of Calculus (Part I) As previously demonstrated (and proven, I hope), we now state it: Fundamental Theorem of Calculus, Part I If f is continuous on [a, b], then the function  EMBED Equation.DSMT4  has a derivative at every point x  [a, b], and  EMBED Equation.DSMT4  2) Examples  Applying the Fundamental Theorem of Calculus (Part I) Straight Forward Homework 5.4.1: Page 302 # 1, 2, 7, 13 Chain Rule 01n Y x  ' 0 4 ; Y \ 4 AB2@APQNP&6Zx|~ξκκκέκ棘 jhHJpr56lm`@ ^@ gd/Oh^hgd/O & Fgd/O`gdMDgd/O^gdMD`  .@BDPRTY$a$gd/Ogd/O p^p`gd/O$%&EFG"+,-gd/O-345TUV Lz@ gdMD^gd6`gd/Ogd6 & Fgd/Ogd/O?C^`npwxQ !!!! ! !"!Ͽ{ljh(h(6EHUjM@R h(h(6UVjh(h(6EHUj-@R h(h(6UVjh(h(6Uh(h(6h(hiQ6h(hiQ5hiQ hiQ5h"M5OJQJ^J h65h)?)@)O)P)T)U)))))****J*Ȼ۰۞ۙێ}}uuuhTthMD5h6jh<~hMDEHUjjI hMDUV hMD6 hMD5jh<~hMDEHUjjI hMDUVjh<~hMDEHUjjI hMDUVjhMDUhMDhmYhMD5h h 5 h 5 hiQ5 h(h +&&G'H'p'q'''''''''((2(3(o(((((*h$d%d&d'dNOPQ^hgdMD & F gdMD & FgdMDgdMD($)%)A)B)C)c)d)))**R*T**Z+\+)$$d%d&d'dNOPQa$gdMDgdMDh^hgdMD-$h$d%d&d'dNOPQ^ha$gdMDJ*L*N*P*T*** .O.P....//1/00002040600000000000112\J\\f]h]]]۪ےրxvqj hehMD hMDH*UhWkqh65 h65j&h<~hMDEHUj jI hMDUVj#h<~hMDEHUj jI hMDUVhmYhMD5h6h  hTthMDhMDhMD56 hMD5hMDjhMDUjK h<~hMDEHUjjI hMDUV)\+,l,,-_--- ...P.Q.`.y.z............ & F gdMDh^hgdMDgdMD & F gdMD................////1/2/_/`///gdMD///8000000000000n1p1111112H\ & F gdMDgdMD)$$d%d&d'dNOPQa$gdMDReversing the limits of integration Homework 5.4.2: Page 302 # 9, 13, 17, 20 3) Recovering (Constructing) a Function Given its Derivative and a Value a) Given: f (x) = sec2x and f(/6) =  EMBED Equation.DSMT4 . Determine f(x). b) Given:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Determine y as a function of x. Homework 5.4.3: Page 302 # 21, 23 4) The Graph of the Function  EMBED Equation.DSMT4  Exploration 1 (page 298) The Effect of Changing a in  EMBED Equation.DSMT4  (Exploration 2 on Page 299) 5) Fundamental Theorem of Calculus (Part II) Integral Evaluation Theorem If f is continuous at every point of [a, b], and if F is any antiderivative of f on [a, b], then  EMBED Equation.DSMT4 . Proof Example Evaluating an Integral Area Connection Analytically Using the Graphing Calculator Using Graphs to Evaluate Definite Integrals (page 304 # 58) Homework 5.4.4: Page 302 # 27, 28, 29, 32, 34, 38, 40, 43, 45, 48, 49, 52, 55, 57, 60, 61 Section 5.5 Trapezoidal Rule The Trapezoid Rule is a method using trapezoids (as opposed to rectangles) to approximate definite integrals and/or area under a curve. 1) See handout Water Pollution Problem 2) Area Formula forH\J\\\\8]:]]]d^e^^^^^^^7_8_9____)$$d%d&d'dNOPQa$gdMDh^hgdMD & F gdMDgdMD]]]]]]^^^^^^$^%^<^=^>^?^e^^^^^^^^^^^^^__________~sf~~~jx6h<~hMDEHUjjI hMDUVh7ehMD5j\3h<~hMDEHUjjI hMDUVhehMD5 hMD5j0h<~hMDEHUjjI hMDUVj-h<~hMDEHUjjI hMDUVj*h<~hMDEHUjTjI hMDUVhMDjhMDU(____```7`G`T`r``` a a a,a-aaaaagd6 & FgdMD & FgdMDgdMD)$$d%d&d'dNOPQa$gdMD_______ ` ` ` `` a a a,a-aDa{a|aaal ll*l0l\llllllºhU|hMD5OJQJ^J hehMDUh(X4hmYhMD5 h/@5hvkhMD5 hMD5jm9h<~hMDEHUjBjI hMDUVjhMDUh7ehMD5hMD a Trapezoid: A = 3) Trapezoid Rule Valid only if all 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