Chapter 5.2, Problem 7E

A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for ${\displaystyle {\int }_{10}^{30}f(x)dx}$.

To determine

The lower estimate and upper estimate for the integral ${\displaystyle {\int }_{10}^{30}f\left(x\right)dx}$.

Answer to Problem 7E

The lower estimate for the integral ${\displaystyle {\int }_{10}^{30}f\left(x\right)dx}$ is $\underset{\_}{-64}$.

The upper estimate for the integral ${\displaystyle {\int }_{10}^{30}f\left(x\right)dx}$ is $\underset{\_}{16}$.

Explanation of Solution

Given information:

The function f is an increasing function.

The number of intervals is $n=5$.

The sub interval value is $\Delta x=4$.

Calculation:

Since the function f is increasing, the Riemann sum $L_5$ provides the lower estimate and $R_5$ provides the upper estimate for the integral ${\displaystyle {\int }_{10}^{30}f\left(x\right)dx}$.

The expression to find the lower estimate is shown below:

$\begin{array}{c}{L}_n={\displaystyle \sum _{i=1}^{5}f\left(x_{i-1}\right)\Delta x}\\ =\Delta x\left[f\left(x_0\right)+f\left(x_1\right)+f\left(x_2\right)+f\left(x_3\right)+f\left(x_4\right)\right]\end{array}$ (1)

Substitute 5 for n, 4 for $\Delta x$, and corresponding x values in Equation (1).

$L_5=4\left[f\left(10\right)+f\left(14\right)+f\left(18\right)+f\left(22\right)+f\left(26\right)\right]$ (2)

Substitute $-12$ for $f\left(10\right)$, $-6$ for $f\left(14\right)$, $-2$ for $f\left(18\right)$, 1 for $f\left(22\right)$, and 3 for $f\left(26\right)$ in Equation (2).

$\begin{array}{c}{L}_5=4\left[f\left(10\right)+f\left(14\right)+f\left(18\right)+f\left(22\right)+f\left(26\right)\right]\\ =4\left(-12-6-2+1+3\right)\\ =-64\end{array}$

Therefore, the lower estimate for the integral ${\displaystyle {\int }_{10}^{30}f\left(x\right)dx}$ is $\underset{\_}{-64}$.

The expression to find the upper estimate is shown below:

$\begin{array}{c}{R}_n={\displaystyle \sum _{i=1}^{5}f\left(x_i\right)\Delta x}\\ =\Delta x\left[f\left(x_1\right)+f\left(x_2\right)+f\left(x_3\right)+f\left(x_4\right)+f\left(x_5\right)\right]\end{array}$ (3)

Substitute 5 for n, 4 for $\Delta x$, and corresponding x values in Equation (1).

$R_5=4\left[f\left(14\right)+f\left(18\right)+f\left(22\right)+f\left(26\right)+f\left(30\right)\right]$ (4)

Substitute $-6$ for $f\left(14\right)$, $-2$ for $f\left(18\right)$, 1 for $f\left(22\right)$, 3 for $f\left(26\right)$, and $8$ for $f\left(30\right)$,in Equation (4).

$\begin{array}{c}{R}_5=4\left[f\left(14\right)+f\left(18\right)+f\left(22\right)+f\left(26\right)+f\left(30\right)\right]\\ =4\left(-6-2+1+3+8\right)\\ =16\end{array}$

Therefore, the upper estimate for the integral ${\displaystyle {\int }_{10}^{30}f\left(x\right)dx}$ is $\underset{\_}{16}$.