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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Using the Trapezoidal Rule and Simpson’s Rule In Exercises 35–40, use the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of the definite integral. Round your answers to four decimal places.

0 4 e x / 2 d x , n = 4

To determine

The value of the integral 04ex2dx,n=4 by using the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of definite integral. Round your answers to four decimal places.

Explanation

Given Information:

The definite integral is 04ex2dx,n=4

Formula used:

1. Trapezoidal Rule:

If a function f is continuous on [a,b], then

02f(x)dx,n=(ba2n)[f(x0)+2f(x0)++2f(xn1)+f(xn)]

2. Simpson’s Rule:

If f is continuous on [a,b] and n is an even integer, then

02f(x)dx,n=(ba3n)[f(x0)+4f(x1)+2f(x2)+4f(x3)++4f(xn1)+f(xn)].

Calculation:

Calculation to get exact value:

Consider the definite integral 04ex2dx,n=4.

04ex2dx=[2ex2]04

Simplify as:

04ex2dx=2e21.7293

Calculation by Trapezoidal Rule:

Consider the definite integral 04ex2dx,n=4.

When, n=4 the width of each subinterval is:

404=44=1

And the end points of subintervals are,

x0=0x1=0+1=1

And,

x2=1+1=2x3=2+1=3

And,

x4=3+1=4

By using Trapezoidal Rule:

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