BuyFind

Single Variable Calculus: Concepts...

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

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

Answer to Problem 3E

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:

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5.2, Problem 3E

Hence,

The value of Riemann sum is 2.322986.

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