Explanation Given: The provided integral is: ∫ 0 8 x 3 d x And the value n = 8 . Formula used: Trapezoidal rule is: ∫ a b f ( x ) d x ≈ b − a 2 n [ f ( x 0 ) + 2 f ( x 1 ) + .... + 2 f ( x n − 1 ) + f ( x n ) ] Simpson’s rule is: ∫ a b p ( x ) d x ≈ ( b − a 3 n ) [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + ...2 f ( x n − 2 ) + 4 f ( x n − 1 ) + f ( x n ) ] The power rule for integration is: ∫ x n d x = x n + 1 n + 1 + c , n ≠ − 1 Calculation: Consider f ( x ) = x 3 . From the provided integral b = 8 , a = 0 and n = 8 . So, the width is: Δ x = b − a n = 8 − 0 8 = 8 8 = 1 Use trapezoidal rule: ∫ 0 8 x 3 d x ≈ 8 − 0 2 ( 8 ) [ f ( 0 ) + 2 f ( 0 + 1 ) + 2 f ( 0 + 1 × 2 ) + 2 f ( 0 + 1 × 3 ) + 2 f ( 0 + 1 × 4 ) + 2 f ( 0 + 1 × 5 ) + 2 f ( 0 + 1 × 6 ) + 2 f ( 0 + 1 × 7 ) + f ( 0 + 1 × 8 ) ] = 8 16 [ f ( 0 ) + 2 f ( 1 ) + 2 f ( 2 ) + 2 f ( 3 ) + 2 f ( 4 ) + 2 f ( 5 ) + 2 f ( 6 ) + 2 f ( 7 ) + f ( 8 ) ] = 1 2 [ ( 0 ) 1 / 3 + 2 ( 1 ) 1 / 3 + 2 ( 2 ) 1 / 3 + 2 ( 3 ) 1 / 3 + 2 ( 4 ) 1 / 3 + 2 ( 5 ) 1 / 3 + 2 ( 6 ) 1 / 3 + 2 ( 7 ) 1 / 3 + ( 8 ) 1 / 3 ] = 1 2 [ 0 + 2 × 1 + 2 × 1.2599 + 2 × 1.4422 + 2 × 1.5874 + 2 × 1.7099 + 2 × 1.8171 + 2 × 1.9129 + 2 ] Simplify the above equation: ∫ 0 8 x 3 d x = 1 2 [ 0 + 2 × 1 + 2 × 1

Calculus Of A Single Variable With Calcchat And Calcview, 11e

11th Edition

ISBN: 9781337275583

Author: Larson

Publisher: Cengage Learning

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Chapter 8.6, Problem 8E

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To calculate: The approximate value of definite integral ∫08x3dx for given value of n by Trapezoidal rule and Simpson’s rule and compare it with the exact value of the definite integral.

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Chapter 8.6, Problem 7E

Chapter 8.6, Problem 9E

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Calculus Of A Single Variable With Calcchat And Calcview, 11e

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The approximate value of definite integral ∫ 0 8 x 3 d x for given value of n by Trapezoidal rule and Simpson’s rule and compare it with the exact value of the definite integral.

Solution Summary: The author compares Trapezoidal rule and Simpson's rule to calculate the approximate value of the definite integral.

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To calculate: The approximate value of definite integral ∫08x3dx for given value of n by Trapezoidal rule and Simpson’s rule and compare it with the exact value of the definite integral.

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Chapter 8 Solutions