Chapter 5.2, Problem 9E

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.

${\displaystyle {\int }_0^8\sin \sqrt{x}dx},n=4$

To determine

To evaluate: The integral usingmidpoint rule.

Answer to Problem 9E

The value of integral ${\displaystyle {\int }_0^8\sin \sqrt{x}dx}$ is $\underset{\_}{6.1820}$.

Explanation of Solution

Given information:

The integral function ${\displaystyle {\int }_0^8\sin \sqrt{x}dx}$ and $n=4$.

Calculation:

Apply Midpoint Rule.

$\begin{array}{c}{\displaystyle {\int }_a^{b}f\left(x\right)dx}\approx {\displaystyle \sum _{i=1}^{n}f\left({\overline{x}}_i\right)\Delta x}\\ =\Delta x\left[f\left({\overline{x}}_1\right)+\mathrm{...}+f\left({\overline{x}}_n\right)\right]\end{array}$ (1)

Find the width $\left(\Delta x\right)$ using the relation:

$\Delta x=\frac{b-a}{n}$

Substitute 8 for b, 0 for a and 4 for $n$ in Equation (1).

$\begin{array}{c}\Delta x=\frac{8-0}{4}\\ =\frac{8}{4}\\ =2\end{array}$

Calculate right end points $x_i$ using the relation:

$x_i=a+i\Delta x$ (2)

Calculate left end points $x_{i-1}$ using Equation (2)

Substitute $\left(i-1\right)$ for $i$ in Equation (2)

$x_{i-1}=a+\left(i-1\right)\Delta x$ (3)

Calculate mid points using the relation:

${\overline{x}}_i=\frac{1}{2}\left(x_{i-1}+x_i\right)$

Substitute $\left(a+i\Delta x\right)$ for $x_i$ and $a+\left(i-1\right)\Delta x$ for $x_{i-1}$

$\begin{array}{c}{\overline{x}}_i=\frac{1}{2}\left(x_{i-1}+x_i\right)\\ =\frac{1}{2}\left[a+\left(i-1\right)\Delta x+a+i\Delta x\right]\\ =\frac{1}{2}\left(a+i\Delta x-\Delta x+a+i\Delta x\right)\\ =\frac{1}{2}\times \left[2\left(a+i\Delta x\right)-\Delta x\right]\end{array}$

${\overline{x}}_i=\left(a+i\Delta x\right)-\frac{\Delta x}{2}$ (4)

Calculate ${\overline{x}}_1$ using Equation (4)

Substitute 0 for $a$, 1 for $i$,and 2 for $\Delta x$ in Equation (4).

$\begin{array}{c}{\overline{x}}_i=\left(a+i\Delta x\right)-\frac{\Delta x}{2}\\ =(0+1\times 2)-\frac{2}{2}\\ =2-1\\ =1\end{array}$

Calculate ${\overline{x}}_2$ using Equation (4).

Substitute 0 for $a$, 2 for $i$,and 2 for $\Delta x$ in Equation (4).

$\begin{array}{c}{\overline{x}}_i=\left(a+i\Delta x\right)-\frac{\Delta x}{2}\\ =(0+2\times 2)-\frac{2}{2}\\ =4-1\\ =3\end{array}$

Calculate ${\overline{x}}_3$ using Equation (4).

Substitute 0 for $a$, 3 for $i$,and 2 for $\Delta x$ in Equation (4)

$\begin{array}{c}{\overline{x}}_i=\left(a+i\Delta x\right)-\frac{\Delta x}{2}\\ =(0+3\times 2)-\frac{2}{2}\\ =6-1\\ =5\end{array}$

Calculate ${\overline{x}}_4$ using Equation (4)

Substitute 0 for $a$, 3 for $i$,and 2 for $\Delta x$ in Equation (4)

$\begin{array}{c}{\overline{x}}_i=\left(a+i\Delta x\right)-\frac{\Delta x}{2}\\ =(0+4\times 2)-\frac{2}{2}\\ =8-1\\ =7\end{array}$

Compare the integral function ${\displaystyle {\int }_0^8\sin \sqrt{x}dx}$ with Equation (1).

$f\left(x\right)=\sin \sqrt{x}$

Calculate $f\left(\overline{x_i}\right)$.

Substitute ${\overline{x}}_i$ for $x$.

$f\left({\overline{x}}_i\right)=\sin \sqrt{{\overline{x}}_i}$ (5)

Calculate $f\left({\overline{x}}_1\right)$ using Equation (5).

Substitute 1 for ${\overline{x}}_1$ in the Equation (5)

$\begin{array}{c}f\left({\overline{x}}_1\right)=\sin \sqrt{{\overline{x}}_i}\\ =\sin \sqrt{\left(1\right)}\\ =0.8415\end{array}$

Calculate $f\left({\overline{x}}_2\right)$ using Equation (5).

Substitute 3 for ${\overline{x}}_2$ in the Equation (5)

$\begin{array}{c}f\left({\overline{x}}_2\right)=\sin \sqrt{{\overline{x}}_2}\\ =\sin \sqrt{\left(3\right)}\\ =0.9870\end{array}$

Calculate $f\left({\overline{x}}_3\right)$ using Equation (5).

Substitute 5 for ${\overline{x}}_3$ in the Equation (5).

$\begin{array}{c}f\left({\overline{x}}_3\right)=\sin \sqrt{{\overline{x}}_3}\\ =\sqrt{\sin \left(5\right)}\\ =0.7867\end{array}$

Calculate $f\left({\overline{x}}_4\right)$ using the Equation (5).

Substitute 7 for ${\overline{x}}_4$ in the Equation (5).

$\begin{array}{c}f\left({\overline{x}}_4\right)=\sin \sqrt{{\overline{x}}_4}\\ =\sin \sqrt{\left(7\right)}\\ =0.4758\end{array}$

Calculate ${\displaystyle {\int }_0^8\sin \sqrt{x}dx}$ using Equation (1):

Substitute $\sin \sqrt{x}$ for $f\left(x\right)$, 2 for $\Delta x$,$0.8415$ for $f\left({\overline{x}}_1\right)$, $0.9870$ for $f\left({\overline{x}}_2\right)$, $0.7867$ for $f\left({\overline{x}}_3\right)$,$n=4$ and $0.4758$ for $f\left({\overline{x}}_4\right)$, $0$ for $a$ and $8$ for $b$ in Equation (1).

$\begin{array}{c}{\displaystyle {\int }_0^8\sin \sqrt{x}dx}={\displaystyle \sum _{i=1}^{4}f\left({\overline{x}}_i\right)\Delta x}\\ =\Delta x\left[f\left({\overline{x}}_1\right)+f\left({\overline{x}}_2\right)+f\left({\overline{x}}_3\right)+f\left({\overline{x}}_4\right)\right]\\ =2\left[0.8415+0.9870+0.7867+0.4758\right]\\ =2\times 3.0910\end{array}$

${\displaystyle {\int }_0^8\sin \sqrt{x}dx}=6.1820$

Thus, the value of ${\displaystyle {\int }_0^8\sin \sqrt{x}dx}$ is 6.1820.