# The integral ∫ 0 2 f ( x ) d x by interpreting it in terms of area.

### 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 31E

(a)

To determine

## To evaluate: The integral ∫02f(x)dx by interpreting it in terms of area.

Expert Solution

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

### Explanation of Solution

The graph of the function f(x) for the interpretation of the area of 02f(x)dx is as shown in Figure 1.

Refer to Figure 1.

The shaded portion represents the area of the trapezoid (A1).

Calculate the value of the integral 02f(x)dx as shown below.

02f(x)dx=A1=12×(a1+b1)h1 (1)

Here, the base of trapezoid is a1 and b1 and the height of the trapezoid is h1,

Substitute 1 for a1, 3 for b1, and 2 for h1 in Equation (1).

02f(x)dx=12×(1+3)×2=4

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

(b)

To determine

### To evaluate: The integral ∫05f(x)dx by interpreting it in terms of area.

Expert Solution

The value of the integral 05f(x)dx is 10.

### Explanation of Solution

The graph of the function f(x) for the interpretation of the area of 05f(x)dx is as shown in Figure 2.

Refer to Figure 2.

The shaded portion represents the summation of the area of trapezoid A1, area of the rectangle A2, and area of the triangle A3.

Calculate the value of the integral 05f(x)dx as shown below.

05f(x)dx=02f(x)dx+23f(x)dx+35f(x)dx=A1+A2+A3=12(a1+b1)h1+b2h2+12b3h3 (2)

Substitute 1 for a1, 3 for b1, 2 for h1, 1 for b2, 3 for h2, 2 for b3, and 3 for h3. in Equation (2).

05f(x)dx=12(1+3)2+(1×3)+12×2×3=4+3+3=10

Thus, the value of the integral 05f(x)dx is 10.

(c)

To determine

### To evaluate: The integral ∫57f(x)dx by interpreting it in terms of area.

Expert Solution

The value of the integral 57f(x)dx is 3.

### Explanation of Solution

Show the graph for area interpretation of the integral 57f(x)dx as in Figure 3.

Refer to Figure 3.

The area of the shaded portion is the triangle A4.

Calculate the value of integral 57f(x)dx as shown below.

57f(x)dx=A4=12×b4×h4 (3)

Substitute 2 for b4 and 3 for h4 in Equation (3).

57f(x)dx=12×2×(3)=3

Therefore, the value of integral 57f(x)dx is 3.

(d)

To determine

### To evaluate: The integral ∫09f(x)dx by interpreting it in terms of area.

Expert Solution

The value of the integral 09f(x)dx is 2.

### Explanation of Solution

Show the graph for area interpretation of 09f(x)dx as in Figure 5.

Refer to Figure 4.

The area of the shaded portion is the area of the trapezoidal A5.

Calculate the value of integral 09f(x)dx as shown below.

09f(x)dx=05f(x)dx+57f(x)dxA5=05f(x)dx+57f(x)dx[12(a5+b5)h5] (4)

Substitute 10 for 05f(x)dx, 3 for 57f(x)dx, 2 for a5, 3 for b5, and 2 for h5 in Equation (4).

09f(x)dx=103[12(2+3)2]=75=2

Therefore, the value of the integral 09f(x)dx is 2.

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