Copyright © Cengage Learning. All rights reserved. 4 Integration 4.1 4.2 4.3 4.4 4.5 4.6 Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. Antiderivatives and Indefinite Integration 4.1 Objectives Write the general solution of a differential equation. Use indefinite integral notation for antiderivatives. Use basic integration rules to find antiderivatives. Find a particular solution of a differential equation. Copyright © Cengage Learning. All rights reserved.

Antiderivatives The derivative of what is 2x? x2 x2 + 1 x2 – 5 …. x2 + C The derivative of what is sinθ -cosθ -cosθ + C The derivative of what is 2? 2x 2x + 1 2x – 5 …. 2x + C The derivative of what is 4x3 x4 x4 + C

Example 1 – Solving a Differential Equation Find the general solution of the differential equation y' = 2. y = 2x + C

Example 2 – Applying the Basic Integration Rules Find the antiderivatives of 3x. Find the integral of 2 sinx dx

Example 3

Example 4

Basic Integration Rules

Example 6

Example 7 Solve the differential equation with an initial condition F(2) = 4

Example 7 – Finding a Particular Solution Find the general solution of and find the particular solution that satisfies the initial condition F(1) = 0.

Example 7 – Solution So, the particular solution cont’d So, the particular solution in that satisfies F(1) = 0 is

Copyright © Cengage Learning. All rights reserved. 4.1 Summary Objectives Write the general solution of a differential equation. Use indefinite integral notation for antiderivatives. Use basic integration rules to find antiderivatives. Find a particular solution of a differential equation. Copyright © Cengage Learning. All rights reserved.

4.2 Area and Summations Sigma Notation Summation Formulas Approximating the Area of a Plane Region Finding the Area by the Limit Definition

Ex 1 Sigma Notation     ≈ 0.4545489741 i = 1 2 3 4 5 6     k = 5 6 7 8

Ex 2 Use sigma notation to write the sum    

Ex 3 Use the summation formulas to evaluate              

Ex 3 Use the summation formulas to evaluate        

Ex 4 Use summation formulas to rewrite the expression without the summation notation        

Ex 4 Use summation formulas to rewrite the expression without the summation notation        

Ex 6 Use the summation formulas and find the limit          

Ex 6 Use the summation formulas and find the limit          

Ex 6 Use the summation formulas and find the limit          

Ex 5 Use left and right endpoints to estimate the area f(x) = 2x + 3, [0,2], 4 rectangles Draw the picture Make an x,y table Calculate the area X Y ½ 1 3/2 2 3 4 5 6 Δx = (2 – 0)/4 = 0.5 7 Area (rect left endpts) = 0.5(3 + 4 + 5 + 6) = 9 Area (rect rt endpts) = 0.5(4 + 5 + 6 + 7) = 11

Ex 5 Use left and right endpoints to estimate the area [0,1], 4 rectangles Draw the picture Make an x,y table Calculate the area X Y ¼ ½ ¾ 1 √3/4 √3/2 √9/4 Δx = (1 – 0)/4 = 0.25 √3 Area (rect left endpts) = 0.25(0 + √3/4 + √3/2 + √9/4) = 0.8977 Area (rect rt endpts) = 0.25(√3/4 + √3/2 + √9/4 + √3) = 1.3307

4.2 Summary Sigma Notation Summation Formulas Approximating the Area of a Plane Region Finding the Area by the Limit Definition

Copyright © Cengage Learning. All rights reserved. 4.3 Definite Integrals Objectives Find the area under a curve using familiar geometric shapes. Evaluate a definite integral. Copyright © Cengage Learning. All rights reserved.

Example 1 – Evaluating a Definite Integral as a Limit Evaluate the definite integral The function f(x) = 2x is integrable on the interval [–2, 1] because it is continuous on [–2, 1].

Definite Integrals

Example 2 Find the area of the region bounded by the graph of f(x) = 4x – x2 and the x-axis Because f is continuous and nonnegative on the closed interval [0, 4], the area of the region is Also show the buttons on the calculator math and graph…

Example 3 – Areas of Common Geometric Figures Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula. a. b. c. ½ circ = ½ π(2)2 Rect = 2x4 Trap = ½ (3)(2+5)

Properties of Definite Integrals

Example 4 – Evaluating Definite Integrals a. Find the value of Given find the value of

Example 5 – Using the Additive Interval Property -1 1

Example 6 – Evaluation of a Definite Integral Evaluate.

Write an integral to represent the area f(x) = 25 − x2 g(y) = y3

= -1/2 = -1/2 + 2 = 3/2 = -5(2 – 4) = -3/2 = 10 = 4(2 + 2 – 4 + ½) The graph of f consists of line segments, as shown in the figure. Evaluate each definite integral by using geometric formulas. = -1/2 = -1/2 + 2 = 3/2 = -5(2 – 4) = -3/2 = 10 = 4(2 + 2 – 4 + ½) = -21/2 = 2

Area Find the area between the curve and the x-axis from x = -2 to x = 4 for f(x) = x2 – 1   -2 -1 1 4  

Copyright © Cengage Learning. All rights reserved. 4.3 Summary Objectives Find the area under a curve using familiar geometric shapes. Evaluate a definite integral. Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 4.4 The Fundamental Theorem of Calculus Objectives Evaluate a definite integral using the Fundamental Theorem of Calculus. Understand and use the Mean Value Theorem for Integrals. Find the average value of a function over a closed interval. Understand and use the Second Fundamental Theorem of Calculus. Understand and use the Net Change Theorem. Copyright © Cengage Learning. All rights reserved.

The Fundamental Theorem of Calculus The two major branches of calculus: differential calculus and integral calculus. At this point, these two problems might seem unrelated—but there is a very close connection. The connection was discovered independently by Isaac Newton and Gottfried Leibniz and is stated in a theorem that is appropriately called the Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus

Example 1 Evaluate each definite integral.

Example 2 Evaluate ½ 2

Example 3 Find the area of the region bounded by the graph of y = 2x2 – 3x + 2, the x-axis, and the vertical lines x = 0 and x = 2. Vertex: (-b/2a, ) (3/4, 7/8) opens up 2

The Mean Value Theorem for Integrals The area of a region under a curve is greater than the area of an inscribed rectangle and less than the area of a circumscribed rectangle. The Mean Value Theorem for Integrals states that somewhere “between” the inscribed and circumscribed rectangles there is a rectangle whose area is precisely equal to the area of the region under the curve.

The Mean Value Theorem for Integrals The value of f(c) given in the Mean Value Theorem for Integrals is called the average value of f on the interval [a, b].

Average Value of a Function In Figure 4.31 the area of the region under the graph of f is equal to the area of the rectangle whose height is the average value. Figure 4.31

Example 4 – Finding the Average Value of a Function Find the average value of f(x) = 3x2 – 2x on the interval [1, 4]. The average value is given by

Example 6 – The Definite Integral as a Function Evaluate the function You could evaluate five different definite integrals, one for each of the given upper limits. However, it is much simpler to fix x (as a constant) temporarily to obtain

The Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus

Example 7 – Using the Second Fundamental Theorem of Calculus Evaluate Note that is continuous on the entire real line. So, using the Second Fundamental Theorem of Calculus, you can write

Example 8 Find the derivative of each:

Net Change Theorem

Example 9 – Using the Net Change Theorem A chemical flows into a storage tank at a rate of 180 + 3t liters per minute, where 0 ≤ t ≤ 60. Find the amount of the chemical that flows into the tank during the first 20 minutes. Let c(t) be the amount of the chemical in the tank at time t. Then c'(t) represents the rate at which the chemical flows into the tank at time t. So, the amount that flows into the tank during the first 20 minutes is 4200 liters.

Net Change Theorem The velocity of a particle moving along a straight line where s(t) is the position at time t. Then its velocity is v(t) = s'(t) and This definite integral represents the net change in position, or displacement, of the particle.

Net Change Theorem When calculating the total distance traveled by the particle, you must consider the intervals where v(t) ≤ 0 and the intervals where v(t) ≥ 0. When v(t) ≤ 0 the particle moves to the left, and when v(t) ≥ 0, the particle moves to the right. To calculate the total distance traveled, integrate the absolute value of velocity |v(t)|.

Net Change Theorem So, the displacement of a particle and the total distance traveled by a particle over [a, b] can be written as

Example 10 – Solving a Particle Motion Problem A particle is moving along a line so that its velocity is v(t) = t3 – 10t2 + 29t – 20 feet per second at time t. What is the displacement of the particle on the time interval 1 ≤ t ≤ 5? b. What is the total distance traveled by the particle on the time interval 1 ≤ t ≤ 5?

Example 10(a) – Solution By definition, you know that the displacement is So, the particle moves feet to the right.

Ex 10(b) – Solution To find the total distance traveled, calculate v(t) = t3 – 10t2 + 29t – 20 cont’d To find the total distance traveled, calculate v(t) = (t – 1)(t – 4)(t – 5), you can determine that v(t) ≥ 0 on [1, 4] and on v(t) ≤ 0 on [4, 5].

Copyright © Cengage Learning. All rights reserved. 4.4 Summary Objectives Evaluate a definite integral using the Fundamental Theorem of Calculus. Understand and use the Mean Value Theorem for Integrals. Find the average value of a function over a closed interval. Understand and use the Second Fundamental Theorem of Calculus. Understand and use the Net Change Theorem. Copyright © Cengage Learning. All rights reserved.

4.5 Integration by Substitution Objectives Use a change of variables to find an indefinite integral. Use the General Power Rule for Integration to find an indefinite integral. Use a change of variables to evaluate a definite integral. Evaluate a definite integral involving an even or odd function. Copyright © Cengage Learning. All rights reserved.

Pattern Recognition In this section you will study techniques for integrating composite functions. You will be using a technique with u-substitution. The role of substitution in integration is comparable to the role of the Chain Rule in differentiation.

Example 1 & 2 Find u = x2 + 1 du = 2x dx Find

Example 3 – Multiplying and Dividing by a Constant Find

Example 4 Find

Example 5 Find

Example 6 Find

Example 7 – Substitution and the General Power Rule

Example 7 – Substitution and the General Power Rule cont’d

Change of Variables for Definite Integrals When using u-substitution with a definite integral, it is often convenient to determine the limits of integration for the variable u rather than to convert the antiderivative back to the variable x and evaluate at the original limits. This change of variables is stated explicitly in the next theorem.

Example 8 – Change of Variables Evaluate Before substituting, determine the new upper and lower limits of integration.

Ex: 9

Integration of Even and Odd Functions Even with a change of variables, integration can be difficult. Occasionally, you can simplify the evaluation of a definite integral over an interval that is symmetric about the y-axis or about the origin by recognizing the integrand to be an even or odd function

Integration of Even and Odd Functions

Example 10 – Integration of an Odd Function Evaluate f(x) = sin3x cos x + sin x cos x f(–x) = sin3(–x) cos (–x) + sin (–x) cos (–x) = –sin3x cos x – sin x cos x = –f(x) Which means that f(x) is odd …

Copyright © Cengage Learning. All rights reserved. 4.5 Summary Objectives Use a change of variables to find an indefinite integral. Use the General Power Rule for Integration to find an indefinite integral. Use a change of variables to evaluate a definite integral. Evaluate a definite integral involving an even or odd function. Copyright © Cengage Learning. All rights reserved.

4.6 Numerical Integration Area under a curve by estimation methods: Left Rectangles Right Rectangles Trapezoids Simpsons Rule Area under a curve EXACT     Copyright © Cengage Learning. All rights reserved.

Ex 1 Estimate w/ left & right rectangles, trap, and simpsons rule Ex 1 Estimate w/ left & right rectangles, trap, and simpsons rule. Then find exact answer f(x) = 2x + 3, [0,2], 4 rectangles Draw the picture Make an x,y table Δx = (2 – 0)/4 = 0.5 X Y ½ 1 3/2 2 Calculate the area 3 4 5 6 7   Area (rect left endpts) = 0.5(3 + 4 + 5 + 6) = 9   Area (rect rt endpts) = 0.5(4 + 5 + 6 + 7) = 11   Area (trapezoids) = ½ * 0.5(3 + 2*4 + 2*5 + 2*6 + 7) = 10   Area (Simpsons) = 1/3 * 0.5(3 + 4*4 + 2*5 + 4*6 + 7) = 10

Ex 2 Estimate w/ left & right rectangles, trap, and simpsons rule Ex 2 Estimate w/ left & right rectangles, trap, and simpsons rule. Then find exact answer [0,1], 4 rectangles     Δx = (1 – 0)/4 = 0.25   X Y ¼ ½ ¾ 1   √3/4   √3/2 √9/4   √3   Area (rect rt endpts) = 0.25(√3/4 + √3/2 + √9/4 + √3) = 1.3307 Area (rect left endpts) = 0.25(0 + √3/4 + √3/2 + √9/4) = 0.8977 Area (trapezoids) = 0.5*0.25(0 + 2*√3/4 + 2*√3/2 + 2*√9/4 + √3) = 1.1142 Area (Simpson’s) = (1/3)*0.25(0 + 4*√3/4 + 2*√3/2 + 4*√9/4 + √3) = 1.1372

On the test you will be given:    

The Trapezoidal Rule One way to approximate a definite integral is to use n trapezoids, as shown in Figure 4.42. In the development of this method, assume that f is continuous and positive on the interval [a, b]. So, the definite integral represents the area of the region bounded by the graph of f and the x-axis, from x = a to x = b. Figure 4.42

The Trapezoidal Rule The area of the ith trapezoid is This implies that the sum of the areas of the n trapezoids

The Trapezoidal Rule Letting you can take the limits as to obtain The result is summarized in the following theorem.

The Trapezoidal Rule

Example 1 – Approximation with the Trapezoidal Rule Use the Trapezoidal Rule to approximate Compare the results for n = 4 and n = 8, as shown in Figure 4.44. Figure 4.44

Example 1 – Solution When n = 4, ∆x = π/4, and you obtain

Example 1 – Solution cont’d When and you obtain

Simpson’s Rule One way to view the trapezoidal approximation of a definite integral is to say that on each subinterval you approximate f by a first-degree polynomial. In Simpson’s Rule, named after the English mathematician Thomas Simpson (1710–1761), you take this procedure one step further and approximate f by second-degree polynomials. Before presenting Simpson’s Rule, we list a theorem for evaluating integrals of polynomials of degree 2 (or less).

Simpson’s Rule

Simpson’s Rule To develop Simpson’s Rule for approximating a definite integral, you again partition the interval [a, b] into n subintervals, each of width ∆x = (b – a)/n. This time, however, n required to be even, and the subintervals are grouped in pairs such that On each (double) subinterval [xi – 2, xi ], you can approximate f by a polynomial p of degree less than or equal to 2.

Simpson’s Rule For example, on the subinterval [x0, x2], choose the polynomial of least degree passing through the points (x0, y0), (x1, y1), and (x2, y2), as shown in Figure 4.45. Figure 4.45

Simpson’s Rule Now, using p as an approximation of f on this subinterval, you have, by Theorem 4.18, Repeating this procedure on the entire interval [a, b] produces the following theorem.

Simpson’s Rule

Example 2 – Solution Use Simpson’s Rule to approximate for n = 4 and n = 8 When n = 4, you have when n = 8, you have

Error Analysis If you must use an approximation technique, it is important to know how accurate you can expect the approximation to be. The following theorem, gives the formulas for estimating the errors involved in the use of Simpson’s Rule and the Trapezoidal Rule. In general, when using an approximation, you can think of the error E as the difference between and the approximation.

Error Analysis

Example 3 – The Approximate Error in the Trapezoidal Rule Determine a value of n such that the Trapezoidal Rule will approximate the value of with an error that is less than or equal to 0.01. Solution: Begin by letting and finding the second derivative of f. The maximum value of |f''(x)| on the interval [0, 1] is |f''(0)| = 1.

Example 3 – Solution So, by Theorem 4.20, you can write cont’d So, by Theorem 4.20, you can write To obtain an error E that is less than 0.01, you must choose n such that 1/(12n2) ≤ 1/100.

Example 3 – Solution cont’d So, you can choose n = 3 (because n must be greater than or equal to 2.89) and apply the Trapezoidal Rule, as shown in Figure 4.46, to obtain Figure 4.46

Example 3 – Solution cont’d So, by adding and subtracting the error from this estimate, you know that