Explanation Given Information: The definite integral is ∫ 0 2 ( 1 + x 2 ) d x , n = 8 Formula used: 1. Trapezoidal Rule: If a function f is continuous on [ a , b ] , then ∫ 0 2 f ( x ) d x , n = ( b − a 2 n ) [ f ( x 0 ) + 2 f ( x 0 ) + ... + 2 f ( x n − 1 ) + f ( x n ) ] 2. Simpson’s Rule: If f is continuous on [ a , b ] and n is an even integer, then ∫ 0 2 f ( x ) d x , n = ( b − a 3 n ) [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + … + 4 f ( x n − 1 ) + f ( x n ) ] . Calculation: Calculation to get exact value: Consider the definite integral ∫ 0 2 ( 1 + x 2 ) d x , n = 8 . ∫ 0 2 ( 1 + x 2 ) d x = [ 1 3 x 3 + x ] 0 2 = [ 1 3 ( 2 ) 3 + 2 − 1 3 ( 0 ) 3 − 0 ] = [ 8 3 + 2 ] Simplify as: ∫ 0 2 ( 1 + x 2 ) d x = 8 + 6 3 = 14 3 = 4.6666 Calculation by Trapezoidal Rule: Consider the definite integral ∫ 0 2 ( 1 + x 2 ) d x , n = 8 . When, n = 8 the width of each subinterval is: 2 − 0 8 = 2 8 = 1 4 And the end points of subintervals are, x 0 = 0 x 1 = 0 + 1 4 = 1 4 And x 2 = 1 4 + 1 4 = 1 2 x 3 = 1 2 + 1 4 = 3 4 And x 4 = 3 4 + 1 4 = 1 x 5 = 1 + 1 4 = 5 4 And x 6 = 5 4 + 1 4 = 3 2 x 7 = 3 2 + 1 4 = 7 4 And, x 8 = 7 4 + 1 4 = 2 By using Trapezoidal Rule: ∫ 0 2 ( 1 + x 2 ) d x = 2 − 0 2 × 8 [ ( ( 0 ) 2 + 1 ) + 2 ( 1 + ( 1 4 ) 2 ) + 2 ( 1 + ( 1 2 ) 2 ) + 2 ( 1 + ( 3 4 ) 2 )

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6, Problem 37RE

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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.

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Chapter 6, Problem 36RE

Chapter 6, Problem 38RE

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The value of the integral ∫ 0 2 ( 1 + x 2 ) d x , 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.

Solution Summary: The author compares the exact value of the definite integral with the Trapezoidal Rule and Simpson’s Rule.

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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.

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Chapter 6 Solutions