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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 2 ( x 2 + 1 ) d x , n = 8

To determine

The value of the integral 02(1+x2)dx,n=8 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 02(1+x2)dx,n=8

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 02(1+x2)dx,n=8.

02(1+x2)dx=[13x3+x]02=[13(2)3+213(0)30]=[83+2]

Simplify as:

02(1+x2)dx=8+63=143=4.6666

Calculation by Trapezoidal Rule:

Consider the definite integral 02(1+x2)dx,n=8.

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

208=28=14

And the end points of subintervals are,

x0=0x1=0+14=14

And

x2=14+14=12x3=12+14=34

And

x4=34+14=1x5=1+14=54

And

x6=54+14=32x7=32+14=74

And,

x8=74+14=2

By using Trapezoidal Rule:

02(1+x2)dx=202×8[((0)2+1)+2(1+(14)2)+2(1+(12)2)+2(1+(34)2)

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