# The value of the integral ∫ 0 2 g ( x ) d x .

### Single Variable Calculus: Concepts...

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

### 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

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.

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

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.

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

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.

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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