Chapter 5.2, Problem 31E

(a)

To determine

To evaluate: The integral ${\displaystyle {\int }_0^{2}f\left(x\right)dx}$ by interpreting it in terms of area.

Answer to Problem 31E

The value of the integral ${\displaystyle {\int }_0^{2}f\left(x\right)dx}$ is 4.

Explanation of Solution

The graph of the function $f\left(x\right)$ for the interpretation of the area of ${\displaystyle {\int }_0^{2}f\left(x\right)dx}$ is as shown in Figure 1.

Refer to Figure 1.

The shaded portion represents the area of the trapezoid $\left(A_1\right)$.

Calculate the value of the integral ${\displaystyle {\int }_0^{2}f\left(x\right)dx}$ as shown below.

$\begin{array}{c}{\displaystyle {\int }_0^{2}f\left(x\right)dx}=A_1\\ =\frac{1}{2}\times \left(a_1+b_1\right)h_1\end{array}$ (1)

Here, the base of trapezoid is $a_1$ and $b_1$ and the height of the trapezoid is $h_1$,

Substitute 1 for $a_1$, 3 for $b_1$, and 2 for $h_1$ in Equation (1).

$\begin{array}{c}{\displaystyle {\int }_0^{2}f\left(x\right)dx}=\frac{1}{2}\times \left(1+3\right)\times 2\\ =4\end{array}$

Thus, the value of the integral ${\displaystyle {\int }_0^{2}f\left(x\right)dx}$ is 4.

(b)

To determine

To evaluate: The integral ${\displaystyle {\int }_0^{5}f\left(x\right)dx}$ by interpreting it in terms of area.

Answer to Problem 31E

The value of the integral ${\displaystyle {\int }_0^{5}f\left(x\right)dx}$ is 10.

Explanation of Solution

The graph of the function $f\left(x\right)$ for the interpretation of the area of ${\displaystyle {\int }_0^{5}f\left(x\right)dx}$ is as shown in Figure 2.

Refer to Figure 2.

The shaded portion represents the summation of the area of trapezoid $A_1$, area of the rectangle $A_2$, and area of the triangle $A_3$.

Calculate the value of the integral ${\displaystyle {\int }_0^{5}f\left(x\right)dx}$ as shown below.

$\begin{array}{c}{\displaystyle {\int }_0^{5}f\left(x\right)dx}={\displaystyle {\int }_0^{2}f\left(x\right)dx}+{\displaystyle {\int }_2^{3}f\left(x\right)dx}+{\displaystyle {\int }_3^{5}f\left(x\right)dx}\\ =A_1+A_2+A_3\\ =\frac{1}{2}\left(a_1+b_1\right)h_1+b_2{h}_2+\frac{1}{2}{b}_3{h}_3\end{array}$ (2)

Substitute 1 for $a_1$, 3 for $b_1$, 2 for $h_1$, 1 for $b_2$, 3 for $h_2$, 2 for $b_3$, and 3 for $h_3$. in Equation (2).

$\begin{array}{c}{\displaystyle {\int }_0^{5}f\left(x\right)dx}=\frac{1}{2}\left(1+3\right)2+\left(1\times 3\right)+\frac{1}{2}\times 2\times 3\\ =4+3+3\\ =10\end{array}$

Thus, the value of the integral ${\displaystyle {\int }_0^{5}f\left(x\right)dx}$ is 10.

(c)

To determine

To evaluate: The integral ${\displaystyle {\int }_5^{7}f\left(x\right)dx}$ by interpreting it in terms of area.

Answer to Problem 31E

The value of the integral ${\displaystyle {\int }_5^{7}f\left(x\right)dx}$ is $-3$.

Explanation of Solution

Show the graph for area interpretation of the integral ${\displaystyle {\int }_5^{7}f\left(x\right)dx}$ as in Figure 3.

Refer to Figure 3.

The area of the shaded portion is the triangle $A_4$.

Calculate the value of integral ${\displaystyle {\int }_5^{7}f\left(x\right)dx}$ as shown below.

$\begin{array}{c}{\displaystyle {\int }_5^{7}f\left(x\right)dx}=A_4\\ =\frac{1}{2}\times b_4\times h_4\end{array}$ (3)

Substitute 2 for $b_4$ and $-3$ for $h_4$ in Equation (3).

$\begin{array}{c}{\displaystyle {\int }_5^{7}f\left(x\right)dx}=\frac{1}{2}\times 2\times \left(-3\right)\\ =-3\end{array}$

Therefore, the value of integral ${\displaystyle {\int }_5^{7}f\left(x\right)dx}$ is $-3$.

(d)

To determine

To evaluate: The integral ${\displaystyle {\int }_0^{9}f\left(x\right)dx}$ by interpreting it in terms of area.

Answer to Problem 31E

The value of the integral ${\displaystyle {\int }_0^{9}f\left(x\right)dx}$ is 2.

Explanation of Solution

Show the graph for area interpretation of ${\displaystyle {\int }_0^{9}f\left(x\right)dx}$ as in Figure 5.

Refer to Figure 4.

The area of the shaded portion is the area of the trapezoidal $A_5$.

Calculate the value of integral ${\displaystyle {\int }_0^{9}f\left(x\right)dx}$ as shown below.

$\begin{array}{c}{\displaystyle {\int }_0^{9}f\left(x\right)dx}={\displaystyle {\int }_0^{5}f\left(x\right)dx}+{\displaystyle {\int }_5^{7}f\left(x\right)dx}-A_5\\ ={\displaystyle {\int }_0^{5}f\left(x\right)dx}+{\displaystyle {\int }_5^{7}f\left(x\right)dx}-\left[\frac{1}{2}\left(a_5+b_5\right)h_5\right]\end{array}$ (4)

Substitute 10 for ${\displaystyle {\int }_0^{5}f\left(x\right)dx}$, $-3$ for ${\displaystyle {\int }_5^{7}f\left(x\right)dx}$, 2 for $a_5$, 3 for $b_5$, and 2 for $h_5$ in Equation (4).

$\begin{array}{c}{\displaystyle {\int }_0^{9}f\left(x\right)dx}=10-3-\left[\frac{1}{2}\left(2+3\right)2\right]\\ =7-5\\ =2\end{array}$

Therefore, the value of the integral ${\displaystyle {\int }_0^{9}f\left(x\right)dx}$ is 2.