Chapter 5.2, Problem 14E

With a programmable calculator or computer (see the instructions for Exercise 5.1.9), compute the left and right Riemann sums for the function f(x) = x/(x + 1) on the interval [0, 2] with n = 100. Explain why these estimates show that

$0.8946<{\displaystyle {\int }_0^2\frac{x}{x+1}dx}<0.9081$

To determine

To Calculate: The left and right Riemann sums for the function $f\left(x\right)=\frac{x}{\left(x+1\right)}$ on interval $\left[0,2\right]$ and $n=100$ using the calculator.

To Explain: the reason for the expression of estimate as follows:

$0.8946<{\displaystyle {\int }_0^2\frac{x}{x+1}dx<0.9081}$

Answer to Problem 14E

The value of left Riemann sums for the function $f\left(x\right)=\frac{1}{\left(x+1\right)}$ on interval $\left[0,2\right]$ and $n=100$ using the calculator is $\underset{\_}{0.89469}$.

The value of right Riemann sums for the function $f\left(x\right)=\frac{1}{\left(x+1\right)}$ on interval $\left[0,2\right]$ and $n=100$ using the calculator is $\underset{\_}{0.90139}$.

Explanation of Solution

The value of the function $f\left(x\right)=\frac{x}{\left(x+1\right)}$ is a increasing function. The left Riemann Sum gives the lower estimate and right Riemann Sum gives upper estimate Thus, get the expression as follows:

$0.8946<{\displaystyle {\int }_0^2\frac{x}{x+1}dx<0.9081}$

Given:

The function as $f\left(x\right)=\frac{x}{\left(x+1\right)}$

The region lies between $x=0$ and $x=2$. So the limits are $a=0$ and $b=2$.

Number of rectangles $n=100$.

Calculation:

Show the Equation of the function $f\left(x\right)$ as follows:

$f\left(x\right)=\frac{x}{\left(x+1\right)}$ (1)

Calculate the value of the function $f\left(x\right)$ for different values of x within the interval $\left[0,2\right]$.

Substitute 0 for x in Equation (1).

$\begin{array}{c}f\left(0\right)=\frac{0}{\left(0+1\right)}\\ =0\end{array}$

Substitute 1.5 for x in Equation (1).

$\begin{array}{c}f\left(1.5\right)=\frac{1.5}{\left(1.5+1\right)}\\ =0.6\end{array}$

Substitute 2 for x in Equation (1).

$\begin{array}{c}f\left(2\right)=\frac{2}{\left(2+1\right)}\\ =0.666\end{array}$

The calculated values of x shows that the function $f\left(x\right)=\frac{x}{\left(x+1\right)}$ is aincreasing function within the interval $\left[0,2\right]$.

The expression to find the left Riemann sum $L_n$ as shown below:

$\begin{array}{c}{L}_n={\displaystyle \sum _{i=1}^{n}f\left(x_i\right)}\Delta x\\ =f\left(x_0\right)\Delta x+f\left(x_1\right)\Delta x+\mathrm{...}+f\left(x_{n-1}\right)\Delta x\end{array}$ (2)

Here, the left endpoint height of first rectangle is $f\left(x_0\right)$, the width is $\Delta x$, height of left endpoint of second rectangle is $f\left(x_1\right)$, and left endpoint height of n^th rectangle is $f\left(x_{n-1}\right)$.

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.

Substitute 2 for b, 0 for a and 100 for n in Equation (3).

$\begin{array}{c}\Delta x=\frac{2-0}{100}\\ =0.02\end{array}$

The value of left and right end points within the interval $\left[0,2\right]$ are shown below:

The left endpoints are $x_0=0$, $x_1=0.02$, $x_2=0.04$$x_{98}=1.96$ and $x_{99}=1.98$.

The right endpoints are $x_1=0.02$, $x_2=0.04$, $x_3=0.06$$x_{99}=1.98$ and $x_{100}=2$.

Calculate the left Riemann sum using calculator.

Substitute 0.02 for $\Delta x$, 100 for n, and values of left end points in Equation (2).

$\begin{array}{c}{L}_{100}={\displaystyle \sum _{i=1}^{100}f\left(x_i\right)}\Delta x\\ =f\left(x_0\right)\Delta x+f\left(x_1\right)\Delta x+\mathrm{...}+f\left(x_{99}\right)\Delta x\\ =f\left(0\right)\times 0.02+f\left(0.02\right)\times 0.02+\mathrm{...}+f\left(1.98\right)\times 0.02\\ =0.89469\end{array}$

The value of left Riemann Sum is $\underset{\_}{0.89469}$.

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 right Riemann sum using calculator.

Substitute 0.02 for $\Delta x$, 100 for n, and values of right end points in Equation (4).

$\begin{array}{c}{R}_n={\displaystyle \sum _{i=1}^{100}f\left(x_i\right)}\Delta x\\ =f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+\mathrm{...}+f\left(x_{100}\right)\Delta x\\ =f\left(0.02\right)\times 0.02+f\left(0.04\right)\times 0.02+\mathrm{...}+f\left(2\right)\times 0.02\\ =0.90802\end{array}$

The value of right Riemann Sum is $\underset{\_}{0.90802}$.

Calculate the value of integral ${\displaystyle {\int }_0^2\frac{x}{x+1}dx}$ using a calculator.

${\displaystyle {\int }_0^2\frac{x}{x+1}dx}=0.90139$ (5)

Compare the left and right Riemann Sum value with value of integral ${\displaystyle {\int }_0^2\frac{x}{x+1}dx}$.

Since, the function $f\left(x\right)=\frac{1}{\left(x+1\right)}$ is a increasing function.

The value of left Riemann sum gives a lower estimate.

The value of the right Riemann sum gives a upper estimate.

Thus, expression the left and right Riemann sum as follows:

$0.89469<{\displaystyle {\int }_0^2\frac{x}{x+1}dx}<0.90802$

Thus, the expression for left and right Riemann sum is shown as $0.89469<{\displaystyle {\int }_0^2\frac{x}{x+1}dx}<0.90802$.