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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.

1 2 1 x 3 d x , n = 8

To determine

To calculate: The approximate value of the integral 121x3dx with n=8 by using the Trapezoidal Rule and Simpson’s Rule and compare the results with the exact value of definite integral.

Explanation

Given Information:

The definite integral is 121x3dx,n=8.

Formula used:

According to Trapezoidal Rule if a function f is continuous on [a,b], then,

abf(x)dx=(ba2n)[f(x0)+2f(x1)++2f(xn1)+f(xn)]

According to Simpson’s Rule if f is continuous on [a,b] and n is an even integer, then,

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

Calculation:

Calculation to get exact value:

Consider the definite integral 121x3dx.

121x3dx=[12x2]12=[12(2)2(12(1)2)]=1+48=38

Simplify as:

121x3dx=38=0.3750

Therefore, the exact value of the integral is 0.375.

Calculation by Trapezoidal Rule:

Consider the definite integral 121x3dx.

When n=8, the width of each subinterval is:

218=18

And the end points of subintervals are

x0=1,x1=1+18=98,x2=98+18=108=54,x3=54+18=10+18=118,x4=118+18=128=64=32,x5=32+18=12+18=138,x6=138+18=148=74,x7=74+18=14+18=158,x8=158+18=168=2

By using Trapezoidal Rule:

121x3dx=212×8[112+2(1(98)3)+2(1(54)3)+2(1(118)3)+2(1(32)

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