# To find the Riemann sum.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 5.2, Problem 3E
To determine

## To find the Riemann sum.

Expert Solution

The value of Riemann sum is 2.322986.

### Explanation of Solution

Given information:

The equation is

f(x)=ex2, 0x2 with n=4

Calculation:

The given expression can be obtained as

f(x)=ex2, 0x2

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

Δx=banΔx=204          [since, b=2,a=0,n=4]Δx=24Δx=0.5              [divede numerator and denominator by 4]

First find

f(0.25)=e0.2521.28402520.715974f(0.75)=e0.7522.11700020.11700f(1.25)=e1.2523.49034221.490342f(1.75)=e1.7525.75460223.754602

Therefore,

The Midpoint rule gives

02(ex2)dx=Δx[f(0.25)+f(0.75)+f(1.25)+f(1.75)]

Put the values of Δx=0.5,f(0.25)=0.715974,f(0.75)=0.11700f(1.25)=1.490342 and f(1.75)=3.754602 on the above equation

02(ex2)dx=Δx[f(0.25)+f(0.75)+f(1.25)+f(1.75)]                     =0.5(0.715974+0.117000+1.490342+3.754602)                     =0.5(5.3619440.715974)                     =0.5(4.64597)                     =2.322985

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 Y= button.

Step 3: Enter the expression ex2 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.

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