Chapter 5.2, Problem 3E

To determine

To find the Riemann sum.

Answer to Problem 3E

The value of Riemann sum is 2.322986.

Explanation of Solution

Given information:

The equation is

  $f\left(x\right)=e^x-2,\text{ 0}\le x\le 2$ with $n=4$

Calculation:

The given expression can be obtained as

  $f\left(x\right)=e^x-2,\text{ 0}\le x\le 2$

The endpoints of the 4 subintervals are $0.5,1,1.5,2$

So, the mid points are $0.25,0.75,1.25,1.75$

The width of the subintervals is

  $\begin{array}{l}\Delta x=\frac{b-a}{n}\\ \Delta x=\frac{2-0}{4}\left[\text{since, }b=2,a=0,n=4\right]\\ \Delta x=\frac{2}{4}\\ \Delta x=0.5\left[\text{divede numerator and denominator by 4}\right]\end{array}$

First find

  $\begin{array}{l}f\left(0.25\right)=e^{0.25}-2\approx 1.284025-2\approx -0.715974\\ f\left(0.75\right)=e^{0.75}-2\approx 2.117000-2\approx 0.11700\\ f\left(1.25\right)=e^{1.25}-2\approx 3.490342-2\approx 1.490342\\ f\left(1.75\right)=e^{1.75}-2\approx 5.754602-2\approx 3.754602\end{array}$

Therefore,

The Midpoint rule gives

  ${\int }_0^2\left(e^x-2\right)dx=\Delta x\left[f\left(0.25\right)+f\left(0.75\right)+f\left(1.25\right)+f\left(1.75\right)\right]$

Put the values of $\begin{array}{l}\Delta x=0.5,f\left(0.25\right)=-0.715974,f\left(0.75\right)=0.11700\\ f\left(1.25\right)=1.490342\text{ and }f\left(1.75\right)=3.754602\end{array}$ on the above equation

  $\begin{array}{l}{\int }_0^2\left(e^x-2\right)dx=\Delta x\left[f\left(0.25\right)+f\left(0.75\right)+f\left(1.25\right)+f\left(1.75\right)\right]\\ \text{ =0}\text{.5}\left(-0.715974+0.117000+1.490342+3.754602\right)\\ \text{ =0}\text{.5}\left(5.361944-0.715974\right)\\ \text{ =0}\text{.5}\left(4.64597\right)\\ \text{ =2}\text{.322985}\end{array}$

The graph is plotted in the below

Graph : Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $e^x-2$ which is required to graph.

Step 4: Press GRAPH button to graph the function and adjust the windows according to the graph.

The graph is obtained as:

  

Hence,

The value of Riemann sum is 2.322986.