(a) Explanation Given: The provided integral is ∫ 0 1 e x 3 d x The value of n for estimating the error is n = 4 Formula used: The error E in approximating ∫ a b f ( x ) d x by the Trapezoidal rule is given by | E | ≤ ( b − a ) 3 12 n 2 [ max | f ' ' ( x ) | ] , a ≤ x ≤ b Calculation: Consider the provided integral ∫ 0 1 e x 3 d x Now, if f has a continuous second derivative on [ a , b ] , then the error E in approximating ∫ a b f ( x ) d x by the Trapezoidal rule is given by | E | ≤ ( b − a ) 3 12 n 2 [ max | f ' ' ( x ) | ] , a ≤ x ≤ b So, begin by letting f ( x ) = e x 3 and finding the second derivative of f Thus, f ' ( x ) = 3 x 2 e x 3 f ' ' ( x ) = 3 ( 2 x + 3 x 4 ) e x 3 Thus, f ' ' ( x ) is continuous on the interval [ 0 , 1 ] as it is defined for all values of x in the interval [ 0 , 1 ] so there are no break points (b) To determine To calculate: Errors in approximating the integral ∫ 0 1 e x 3 d x using Simpson’s rule for n = 4 .

Calculus: Early Transcendental Functions

7th Edition

ISBN: 9781337552516

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 8.6, Problem 28E

(a)

To determine

To calculate: Errors in approximating the integral $∫01ex3dx$ using Trapezoidal rule for $n=4$ .

(b)

To determine

To calculate: Errors in approximating the integral $∫01ex3dx$ using Simpson’s rule for $n=4$ .

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Chapter 8.6, Problem 27E

Chapter 8.6, Problem 29E

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Calculus: Early Transcendental Functions

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Errors in approximating the integral ∫ 0 1 e x 3 d x using Trapezoidal rule for n = 4 .

Solution Summary: The author calculates the error in approximating the integral by trapezoidal rule for n=4.