Chapter 5.2, Problem 15E

To determine

To Calculate: The right Riemann sums $R_n$ for the integral ${\displaystyle {\int }_0^{\pi }\sin x}dx$ for $n=5,10,50,and100$ using the calculator.

To Find: The values of the numbers appear to be approaching.

Answer to Problem 15E

The value of right Riemann sums for the integral ${\displaystyle {\int }_0^{\pi }\sin x}dx$ for $n=5,10,50,and100$ is tabulated in table 1.

The value in the Table 1 is approaching to a value of 2.

Explanation of Solution

Given:

The integral function as ${\displaystyle {\int }_0^{\pi }\sin x}dx$

Number of rectangles $n=5,10,50\text{and}100$.

Calculation:

Show the Equation of the integral as follows:

${\displaystyle {\int }_0^{\pi }\sin x}dx$ (1)

Consider the value of the function $f\left(x\right)=\sin x$ (2)

For $n=5$

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

$\Delta x=\frac{b-a}{n}$ (3)

Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.

The limits of the integral ${\displaystyle {\int }_0^{\pi }\sin x}dx$, $a=0$ and $b=\pi$.

Substitute $\pi$ for b, 5 for n, and 0 for a in Equation (3).

$\begin{array}{c}\Delta x=\frac{\pi -0}{5}\\ =\frac{\pi }{5}\end{array}$

The right endpoints are $x_1=\frac{\pi }{5}$, $x_2=\frac{2\pi }{5}$, $x_3=\frac{3\pi }{5}$, $x_4=\frac{4\pi }{5}$ and $x_5=\pi$.

The expression to find the right Riemann sum $R_n$ as shown below:

$\begin{array}{c}{R}_n={\displaystyle \sum _{i=1}^{n}f\left(x_i\right)}\Delta x\\ =f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+\mathrm{...}+f\left(x_n\right)\Delta x\end{array}$ (4)

Here, the right endpoint height of first rectangle is $f\left(x_1\right)$, the width is $\Delta x$, height of right endpoint of second rectangle is $f\left(x_2\right)$, and left endpoint height of n^th rectangle is $f\left(x_n\right)$.

Calculate the value of $f\left(x_1\right)$ using the Equation (2).

Substitute $\frac{\pi }{5}$ for $x_1$ in Equation (2).

$\begin{array}{c}f\left(x_1\right)=\sin x_1\\ =\sin \left(\frac{\pi }{5}\right)\\ =0.0109\end{array}$

Calculate the value of $f\left(x_2\right)$ using the Equation (2).

Substitute $\frac{2\pi }{5}$ for $x_2$ in Equation (2).

$\begin{array}{c}f\left(x_2\right)=\sin x_2\\ =\sin \left(\frac{2\pi }{5}\right)\\ =0.02193\end{array}$

Similarly calculate the value of $f\left(x\right)$ for remaining values of x.

Calculate the right Riemann sum using calculator.

Substitute $\frac{\pi }{5}$ for $\Delta x$, 5 for n, and values of right end points in Equation (4).

$\begin{array}{c}{R}_n={\displaystyle \sum _{i=1}^{5}f\left(x_i\right)}\Delta x\\ =f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+\mathrm{...}+f\left(x_5\right)\Delta x\\ =f\left(\frac{\pi }{5}\right)\times \frac{\pi }{5}+f\left(\frac{2\pi }{5}\right)\times \frac{\pi }{5}+\mathrm{...}+f\left(\pi \right)\times \frac{\pi }{5}\\ =1.933766\end{array}$

The value of right Riemann Sum for $n=5$ is $\underset{\_}{1.933766}$.

For $n=10$

Find the width $\left(\Delta x\right)$ using the Equation (3):

Substitute $\pi$ for b , 10 for n, and 0 for a in Equation (3).

$\begin{array}{c}\Delta x=\frac{\pi -0}{10}\\ =\frac{\pi }{10}\end{array}$

The right endpoints are $x_1=\frac{\pi }{10}$, $x_2=\frac{2\pi }{10}$, $x_3=\frac{3\pi }{10}$$x_{10}=\pi$.

Calculate the value of $f\left(x_1\right)$ using the Equation (2).

Substitute $\frac{\pi }{10}$ for $x_1$ in Equation (2).

$\begin{array}{c}f\left(x_1\right)=\sin x_1\\ =\sin \left(\frac{\pi }{10}\right)\\ =5.483\times {10}^{-3}\end{array}$

Calculate the value of $f\left(x_2\right)$ using the Equation (2).

Substitute $\frac{2\pi }{10}$ for $x_2$ in Equation (2).

$\begin{array}{c}f\left(x_2\right)=\sin x_2\\ =\sin \left(\frac{2\pi }{10}\right)\\ =0.01096\end{array}$

Similarly calculate the value of $f\left(x\right)$ for remaining values of x.

Calculate the right Riemann sum using calculator.

Substitute $\frac{\pi }{10}$ for $\Delta x$, 10 for n, and values of right end points in Equation (4).

$\begin{array}{c}{R}_{10}={\displaystyle \sum _{i=1}^{10}f\left(x_i\right)}\Delta x\\ =f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+\mathrm{...}+f\left(x_{10}\right)\Delta x\\ =f\left(\frac{\pi }{10}\right)\times \frac{\pi }{10}+f\left(\frac{2\pi }{10}\right)\times \frac{\pi }{10}+\mathrm{...}+f\left(\pi \right)\times \frac{\pi }{10}\\ =1.983524\end{array}$

The value of right Riemann Sum for $n=10$ is $\underset{\_}{1.983524}$.

For $n=50$

Find the width $\left(\Delta x\right)$ using the Equation (3):

Substitute $\pi$ for b , 50 for n, and 0 for a in Equation (3).

$\begin{array}{c}\Delta x=\frac{\pi -0}{50}\\ =\frac{\pi }{50}\end{array}$

The right endpoints are $x_1=\frac{\pi }{50}$, $x_2=\frac{2\pi }{50}$, $x_3=\frac{3\pi }{50}$$x_{50}=\pi$.

Calculate the value of $f\left(x_1\right)$ using the Equation (2).

Substitute $\frac{\pi }{50}$ for $x_1$ in Equation (2).

$\begin{array}{c}f\left(x_1\right)=\sin x_1\\ =\sin \left(\frac{\pi }{50}\right)\\ =1.0966\times {10}^{-3}\end{array}$

Calculate the value of $f\left(x_2\right)$ using the Equation (2).

Substitute $\frac{2\pi }{50}$ for $x_2$ in Equation (2).

$\begin{array}{c}f\left(x_2\right)=\sin x_2\\ =\sin \left(\frac{2\pi }{50}\right)\\ =2.1932\times {10}^{-3}\end{array}$

Similarly calculate the value of $f\left(x\right)$ for remaining values of x.

Calculate the right Riemann sum using calculator.

Substitute $\frac{\pi }{50}$ for $\Delta x$, 50 for n, and values of right end points in Equation (4).

$\begin{array}{c}{R}_{50}={\displaystyle \sum _{i=1}^{50}f\left(x_i\right)}\Delta x\\ =f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+\mathrm{...}+f\left(x_{50}\right)\Delta x\\ =f\left(\frac{\pi }{50}\right)\times \frac{\pi }{50}+f\left(\frac{2\pi }{50}\right)\times \frac{\pi }{50}+\mathrm{...}+f\left(\pi \right)\times \frac{\pi }{50}\\ =1.999342\end{array}$

The value of right Riemann Sum for $n=50$ is $\underset{\_}{1.999342}$.

For $n=100$

Find the width $\left(\Delta x\right)$ using the Equation (3):

Substitute $\pi$ for b , 100 for n, and 0 for a in Equation (3).

$\begin{array}{c}\Delta x=\frac{\pi -0}{100}\\ =\frac{\pi }{100}\end{array}$

The right endpoints are $x_1=\frac{\pi }{100}$, $x_2=\frac{2\pi }{100}$, $x_3=\frac{3\pi }{100}$$x_{100}=\pi$.

The expression to find the right Riemann sum $R_n$ as shown below:

$\begin{array}{c}{R}_n={\displaystyle \sum _{i=1}^{n}f\left(x_i\right)}\Delta x\\ =f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+\mathrm{...}+f\left(x_n\right)\Delta x\end{array}$ (5)

Here, the right endpoint height of first rectangle is $f\left(x_1\right)$, the width is $\Delta x$, height of right endpoint of second rectangle is $f\left(x_2\right)$, and left endpoint height of n^th rectangle is $f\left(x_n\right)$.

Calculate the value of $f\left(x_1\right)$ using the Equation (2).

Substitute $\frac{\pi }{100}$ for $x_1$ in Equation (2).

$\begin{array}{c}f\left(x_1\right)=\sin x_1\\ =\sin \left(\frac{\pi }{100}\right)\\ =5.483\times {10}^{-4}\end{array}$

Calculate the value of $f\left(x_2\right)$ using the Equation (2).

Substitute $\frac{2\pi }{100}$ for $x_2$ in Equation (2).

$\begin{array}{c}f\left(x_2\right)=\sin x_2\\ =\sin \left(\frac{2\pi }{100}\right)\\ =1.0966\times {10}^{-3}\end{array}$

Similarly calculate the value of $f\left(x\right)$ for remaining values of x.

Calculate the right Riemann sum using calculator.

Substitute $\frac{\pi }{100}$ for $\Delta x$, 100 for n, and values of right end points in Equation (5).

$\begin{array}{c}{R}_{100}={\displaystyle \sum _{i=1}^{100}f\left(x_i\right)}\Delta x\\ =f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+\mathrm{...}+f\left(x_{50}\right)\Delta x\\ =f\left(\frac{\pi }{100}\right)\times \frac{\pi }{100}+f\left(\frac{2\pi }{100}\right)\times \frac{\pi }{100}+\mathrm{...}+f\left(\pi \right)\times \frac{\pi }{100}\\ =1.999836\end{array}$

The value of right Riemann Sum for $n=100$ is $\underset{\_}{1.999836}$.

Tabulate the values of Riemann sum for $n=5,10,50\text{and}100$ as shown in table 1.

n	$R_n$
5	1.933766
10	1.983524
50	1.999342
100	1.999836

Table 1

Refer to Table 1.

The value of the Riemann sum is approaching to 2.