Explanation Given Information: The definite integral is ∫ 0 1 x e 3 x 2 d x , n = 4 Formula used: 1. Trapezoidal Rule: If a function f is continuous on [ a , b ] , then ∫ a b f ( x ) d x ≈ ( 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 ∫ a b f ( x ) d x ≈ ( 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 1 x e 3 x 2 d x , n = 4 . ∫ 0 1 x e 3 x 2 d x = 1 6 [ e 3 x 2 ] 0 1 = 1 6 e 3 − 1 6 Simplify as: ∫ 0 1 x e 3 x 2 d x = e 3 − 1 6 ≈ 3.1809 Therefore, exact value: ∫ 0 1 x e 3 x 2 d x ≈ 3.1809 . Calculation by Trapezoidal Rule: Consider the definite integral ∫ 0 1 x e 3 x 2 d x , n = 4 . When n = 4 , the width of each subinterval is, 1 − 0 4 = 1 4 And the end points of subintervals are, For x 0 , x 0 = 0 For x 1 , x 1 = 1 4 For x 2 , x 2 = 1 4 + 1 4 = 1 2 For x 3 , x 3 = 1 2 + 1 4 = 3 4 For x 4 , x 4 = 3 4 + 1 4

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ISBN: 8220101426222

Author: Larson

Publisher: CENGAGE L

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Chapter 6.3, Problem 9E

To determine

To calculate: The value of the integral ∫01xe3x2dx,n=4 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.

∫01xe3x2dx,n=4.

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Chapter 6.3, Problem 8E

Chapter 6.3, Problem 10E

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Solution Summary: The author compares the results obtained from exact value and Simpson's rule and Trapezoidal Rule.

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Chapter 6 Solutions