Chapter 5: Integration Section 5.1 An Area Problem; A Speed-Distance Problem An Area Problem An Area Problem (continued) Upper Sums and Lower Sums Overview.

Name: Chapter 5: Integration Section 5.1 An Area Problem; A Speed-Distance Problem An Area Problem An Area Problem (continued) Upper Sums and Lower Sums Overview.
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Description: Chapter 5: Integration Section 5.1 An Area Problem; A Speed-Distance Problem An Area Problem An Area Problem (continued) Upper Sums and Lower Sums Overview.

Chapter 5: Integration Section 5.1 An Area Problem; A Speed-Distance Problem An Area Problem An Area Problem (continued) Upper Sums and Lower Sums Overview of the Speed Distance Problem Section 5.2 The Definite Integral of a Continuous Function Partitions Partitions; Upper Sum and Lower Sum Example Definition 5.2.3 Use of "dummy" Variables Integral of a Constant Function Integral of the Identity Function The Integral as the Limit of Riemann Sums Section 5.3 The Function Theorem 5.3.1 The Effects of Adding Points to a Partition Theorem 5.3.2 Properties of Integration Theorem 5.3.5 Integration when f > 0 Section 5.4 The Fundamental Theorem of Integral Calculus Antiderivatives and the Fundamental Theorem of Integral Calculus Some Common Antiderivatives and Examples Linearity of the Integral Section 5.5 Some Area Problems Area of Ω Signed Area Section 5.6 Indefinite Integrals Indefinite integrals Common Indefinite Integrals Linearity Properties Application to Motion Example Section 5.7 Working Back from the Chain Rule; The u-Substitution Theorem 5.7.1 Example Substitution in Definite Integrals Section 5.8 Additional Properties of the Definite Integral Properties I and II Properties III and IV Property V Property VI Section 5.9 Mean-Value Theorems for Integrals; Average Value of a Function First Mean-Value Theorem for Integrals Second Mean-Value Theorem for Integrals Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

An Area Problem; A Speed-Distance Problem
In Figure you can see a region Ω bounded above by the graph of a continuous function f, bounded below by the x-axis, bounded on the left by the line x = a, and bounded on the right by the line x = b. The question before us is this: What number, if any, should be called the area of Ω? To begin to answer this question, we split up the interval [a, b] into a finite number of subintervals [x0, x1], [x1, x2], , [xn−1, xn] with a = x0 < x1 < · · · < xn = b. This breaks up the region Ω into n subregions: Ω1, Ω2, , Ωn. (Figure 5.1.2) We can estimate the total area of Ω by estimating the area of each subregion Ωi and adding up the results. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

A sum of the form m1Δx1 + m2Δx2 +· · ·+mnΔxn (Figure 5.1.4) is called a lower sum for f. A sum of the form M1Δx1 + M2 Δx2 +· · ·+ Mn Δxn (Figure 5.1.5) is called an upper sum for f. For a number to be a candidate for the title “area of Ω,” it must be greater than or equal to every lower sum for f and it must be less than or equal to every upper sum. It can be proven that with f continuous on [a, b] there is one and only one such number. This number we call the area of Ω. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

If an object moves at a constant speed for a given period of time, then the total distance traveled is given by the familiar formula distance = speed × time. Suppose now that during the course of the motion the speed ν does not remain constant; suppose that it varies continuously. How can we calculate the distance traveled in that case? To answer this question, we suppose that the motion begins at time a, ends at time b, and during the time interval [a, b] the speed varies continuously. As in the case of the area problem, we begin by breaking up the interval [a, b] into a finite number of subintervals: [t0, t1], [t1, t2], , [tn−1, tn] with a = t0 < t1 < · · · < tn = b. On each subinterval [ti−1, ti ] the object attains a certain maximum speed Mi and a certain minimum speed mi. The total distance traveled during the full time interval [a, b], call it s, must be the sum of the distances traveled during the subintervals [ti−1, ti ]; thus we must have s = s1 + s2 + · · · + sn. Similar to the area problem it can be shown that s must be greater than or equal to every lower sum for the speed function, and it must be less than or equal to every upper sum. It turns out that there is one and only one such number, and this is the total distance traveled. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Definite Integral of a Continuous Function
Example The sets {0, 1}, {0, ½, 1}, {0, ¼, ½, 1}, {0,1/4, 1/3, 1/2, 5/8, 1} are all partitions of the interval [0, 1]. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

If P = {x0, x1, x2, , xn−1, xn} is a partition of [a, b], then P breaks up [a, b] into n subintervals [x0, x1], [x1, x2], , [xn−1, xn] of lengths Δx1, Δx2, , Δxn. Suppose now that f is continuous on [a, b]. Then on each subinterval [xi−1, xi] the function f takes on a maximum value, Mi , and a minimum value, mi . Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Example The function f (x) = 1 + x2 is continuous on [0, 1]. The partition P = {0, ½, ¾, 1} breaks up [0, 1] into three subintervals [x0, x1] = [0, ½], [x1, x2] = [½, ¾], [x2, x3] = [¾, 1] of lengths Δx1 = ½ − 0 = ½, Δx2 = ¾ − ½ = ¼, Δx3 = 1 − ¾= ¼. Since f increases on [0, 1], it takes on its maximum value at the right endpoint of each subinterval: The minimum values are taken on at the left endpoints: Thus and Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

In the expression the letter x is a “dummy variable”; in other words, it can be replaced by any letter not already in use. Thus, for example, all denote exactly the same quantity, the definite integral of f from a to b. From the introduction to this chapter, you know that if f is nonnegative and continuous on [a, b], then the integral of f from x = a to x = b gives the area below the graph of f from x = a to x = b: Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Integral as the Limit of Riemann Sums
Figure illustrates the idea that the definite integral of a continuous function is the limit of Riemann sums . Here the base interval is broken up into eight subintervals. The point is chosen from [x0, x1], from [x1, x2], and so on. which in expanded form reads Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The integral from any number to itself is defined to be zero:
Until now we have integrated only from left to right: from a number a to a number b greater than a. We integrate in the other direction by defining The integral from any number to itself is defined to be zero: Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

F(x) = area from a to x and F(x + h) = area from a to x + h. Therefore
F(x + h) – F(x) = area from x to x + h. For small h this is approximately f (x) h. Thus is approximately Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

intervals on which F decreases.
Example Set for all real numbers x. (a) Find the critical points of F and determine the intervals on which F increases and the intervals on which F decreases. (b) Determine the concavity of the graph of F and find the points of inflection (if any). (c) Sketch the graph of F. Solution (a) To find the intervals on which F increases and the intervals on which F decreases, we examine the first derivative of F. By Theorem 5.3.5, for all real x. Since F´ (x) > 0 for all real x, F increases on (−∞,∞); there are no critical points. (b) To determine the concavity of the graph and to find the points of inflection, we use the second derivative The sign of F´´ and the behavior of the graph of F are as follows: (c) Since F (0) = 0 and F´ (0) = 1, the graph passes through the origin with slope 1. A sketch of the graph is shown in Figure Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Fundamental Theorem of Integral Calculus

Example Evaluate Solution As an antiderivative for f (x) = x2, we can use the function G(x) = ⅓x3. By the fundamental theorem, NOTE: Any other antiderivative of f (x) = x2 has the form H(x) = ⅓x3 + C for some constant C. Had we chosen such an H instead of G, then we would have had the C’s would have canceled out. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Linearity of the Integral
I. Constants may be factored through the integral sign: II. The integral of a sum is the sum of the integrals: III. The integral of a linear combination is the linear combination of the integrals: Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Linearity of the Integral
Example Evaluate Solution Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Some Area Problems Look at the region Ω shown in Figure The upper boundary of Ω is the graph of a nonnegative function f and the lower boundary is the graph of a nonnegative function g. We can obtain the area of Ω by calculating the area of Ω1 and subtracting off the area of Ω2. Since we have Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Some Area Problems The area between the graph of f and the x-axis from x = a to x = e is the sum area of Ω1 + area of Ω2 + area of Ω3 + area of Ω4 This area is Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Indefinite Integrals Consider a continuous function f . If F is an antiderivative for f on [a, b], then If C is a constant, then Thus we can replace (1) by writing If we have no particular interest in the interval [a, b] but wish instead to emphasize that F is an antiderivative for f , which on open intervals simply means that F´ = f , then we omit the a and the b and simply write Antiderivatives expressed in this manner are called indefinite integrals. The constant C is called the constant of integration; it is an arbitrary constant and we can assign to it any value we choose. Each value of C gives a particular antiderivative, and each antiderivative is obtained from a particular value of C. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Indefinite Integrals The linearity properties of definite integrals also hold for indefinite integrals. Example Calculate Solution (writing C for C1 + C2) Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Indefinite Integrals Application to Motion Example
An object moves along a coordinate line with velocity v(t) = 2 − 3t + t2 units per second. Its initial position (position at time t = 0) is 2 units to the right of the origin. Find the position of the object 4 seconds later. Solution Let x(t) be the position (coordinate) of the object at time t. We are given that x(0) = 2. Since x´(t) = v(t), Since x(0) = 2 and x(0) = 2(0) − 3∕2(0)2 + ⅓(0)3 + C = C, we have C = 2 and x(t) = 2t − 3∕2 t2 + ⅓t3 + 2. The position of the object at time t = 4 is the value of this function at t = 4: x(4) = 2(4) − 3∕2(4)2 + ⅓(4)3 + 2 = 7⅓ At the end of 4 seconds the object is 7⅓ units to the right of the origin. The motion of the object is represented schematically in Figure Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The u-Substitution Example Calculate Solution
Set u = 3 + 5x, du = 5 dx. Then and Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The u-Substitution The Definite Integral
This formula is called the change-of-variables formula. The formula can be used to evaluate provided that u´ is continuous on [a, b] and f is continuous on the set of values taken on by u on [a, b]. Since u is continuous, this set is an interval that contains a and b. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Additional Properties of the Definite Integral
I. The integral of a nonnegative continuous function is nonnegative: The integral of a positive continuous function is positive: II. The integral is order-preserving: for continuous functions f and g, and Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

III. Just as the absolute value of a sum of numbers is less than or equal to the sum of the absolute values of those numbers, |x1 + x2 +· · ·+ xn| ≤ |x1| + |x2|+· · ·+|xn|, the absolute value of an integral of a continuous function is less than or equal to the integral of the absolute value of that function: IV. If f is continuous on [a, b], then where m is the minimum value of f on [a, b] and M is the maximum. Reasoning: m(b − a) is a lower sum for f and M(b − a) is an upper sum. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

V. If f is continuous on [a, b] and u is a differentiable function of x with values in [a, b], then for all u(x)  (a, b) Example Find Solution At this stage you probably cannot carry out the integration: it requires the natural logarithm function. (Not introduced in this text until Chapter 7.) But for our purposes, that doesn’t matter. By (5.8.7), Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Additional Properties
VI. Now a few words about the role of symmetry in integration. Suppose that f is continuous on an interval of the form [−a, a], a closed interval symmetric about the origin. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Mean-Value Theorems for Integrals

Chapter 5: Integration Section 5.1 An Area Problem; A Speed-Distance Problem An Area Problem An Area Problem (continued) Upper Sums and Lower Sums Overview.

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Chapter 5: Integration Section 5.1 An Area Problem; A Speed-Distance Problem An Area Problem An Area Problem (continued) Upper Sums and Lower Sums Overview.

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Presentation on theme: "Chapter 5: Integration Section 5.1 An Area Problem; A Speed-Distance Problem An Area Problem An Area Problem (continued) Upper Sums and Lower Sums Overview."— Presentation transcript:

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