Explanation Given Information: The given definite integral is ∫ 0 1 x 3 d x . Formula used: Simpson’s Rule: If f is continuous on [ a , b ] and n is an even integer, then ∫ a b 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 ) ] . The errors E in approximating ∫ a b f ( x ) d x by the Simpson’s Rule: | E | ≤ ( b − a ) 5 180 n 2 [ max | f 4 ( x ) | ] , a ≤ x ≤ b Proof: Consider the definite integral ∫ 0 1 x 3 d x . To prove Simpson’s Rule is exact when approximating the integral of a cubic polynomial function. Let P 3 ( x ) = a x 3 + b x 2 + c x + d is a cubic polynomial function. The first derivative of P 3 ( x ) = a x 3 + b x 2 + c x + d is, P 3 1 ( x ) = d d x ( a x 3 + b x 2 + c x + d ) = 3 a x 2 + 2 b x + c The second derivative of the function is, P 3 2 ( x ) = d d x ( 3 a x 2 + 2 b x + c ) = 6 a x + 2 b The third derivative of the function is, P 3 3 <

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6.3, Problem 48E

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To prove: The Simpson’s Rule is exact when approximating the integral of a cubic polynomial function, and also demonstrates the result with n=2 for the function ∫01x3dx.

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Chapter 6.3, Problem 47E

Chapter 6.4, Problem 1CP

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Calculus: An Applied Approach (MindTap Course List)

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To prove: The Simpson’s Rule is exact when approximating the integral of a cubic polynomial function, and also demonstrates the result with n = 2 for the function ∫ 0 1 x 3 d x .

Solution Summary: The author proves Simpson's Rule is exact when approximating the integral of a cubic polynomial function.

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To prove: The Simpson’s Rule is exact when approximating the integral of a cubic polynomial function, and also demonstrates the result with n=2 for the function ∫01x3dx.

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Chapter 6 Solutions