BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 5.2, Problem 32E

(a)

To determine

To evaluate: The value of the integral 02g(x)dx.

Expert Solution

Answer to Problem 32E

The value of the integral 02g(x)dx is 4.

Explanation of Solution

Given:

A graph of function y=f(x).

Calculation:

Show the graph for area interpretation of 02g(x)dx as in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5.2, Problem 32E , additional homework tip  1

Refer to Figure 1.

The area of shaded portion is area of triangle A1.

Calculate the value of the integral 02g(x)dx shown below:

02g(x)dx=A1=12b1h1 (1)

Substitute 2 for b1 and 4 for h1 in Equation (1).

02g(x)dx=12×2×4=4

Thus, the value of the integral 02g(x)dx is 4_.

(b)

To determine

To evaluate: The value of the integral 26g(x)dx.

Expert Solution

Answer to Problem 32E

The value of the integral 26g(x)dx is 2π_.

Explanation of Solution

Given:

A graph of function y=f(x).

Calculation:

Draw the graph for area interpretation of the integral 26g(x)dx as shown in Figure 2.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5.2, Problem 32E , additional homework tip  2

Refer to Figure 2.

The shaded portion represents area of semicircle A2.

Consider that the area lies below x-axis. Hence, A2 is negative value.

Calculate the value of the integral 26g(x)dx as shown below.

26g(x)dx=A2=12πr22 (2)

Substitute 2 for r2 in Equation (2).

26g(x)dx=12π(2)2=2π

Thus, the value of the integral 26g(x)dx is 2π_.

(c)

To determine

The value of the integral 07g(x)dx.

Expert Solution

Answer to Problem 32E

The value of the integral 07g(x)dx is 4.52π_.

Explanation of Solution

Draw the graph for area interpretation of the integral 07g(x)dx as shown in Figure 3.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5.2, Problem 32E , additional homework tip  3

Refer to Figure 3.

The shaded portion represents area of triangle A3.

Calculate the value of the integral 07g(x)dx as shown below.

07g(x)dx=A1+A2+A3=A1+A2+12b3h3 (3)

Substitute 4 for A1, 2π for A2, 1 for b3 and 1 for h3 in Equation (3).

07g(x)dx=42π+12(1)(1)=42π+0.5=4.52π

Thus, the value of the integral 07g(x)dx is 4.52π_.

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