Explanation 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 ) ] Calculation: Consider f ( x ) = x tan x Since n = 4 Therefore, add π / 4 − 0 4 = π 16 to each term to get values of x 0 , x 2 ....... x m where x 0 = 0 and x 4 = π / 4 . Thus, the values of x are: x 1 = 0 + π / 16 = π 16 x 2 = π 16 + π 16 = π 8 And, x 3 = π 8 + π 16 = 3 π 16 x 4 = 3 π 16 + π 16 = π 4 Now, by trapezoidal rule: Here b = π 4 , a = 0 and n = 4 . Therefore, ∫ 0 π / 4 x tan x d x ≈ π / 4 − 0 2 ( 4 ) [ f ( 0 ) + 2 f ( π 16 ) + 2 f ( π 8 ) + 2 f ( 3 π 16 ) + f ( π 4 ) ] = π 32 [ ( 0 ) tan ( 0 ) + 2 ( π 16 ) tan ( π 16 ) + 2 ( π 8 ) t a n ( π 8 ) + 2 ( 3 π 16 ) t a n ( 3 π 16 ) + π 4 t a n ( π 4 ) ] = π 32 [ 0 + 0

Bundle: Calculus: Early Transcendental Functions, 7th + Webassign, Multi-term Printed Access Card

7th Edition

ISBN: 9781337888936

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 8.6, Problem 23E

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To calculate: The approximate value of definite integral ∫0π/4xtanxdx for n=4 by Trapezoidal rule and Simpson’s rule and compare it with the approximate value of the integral using graphing utility.

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Chapter 8.6, Problem 22E

Chapter 8.6, Problem 24E

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Chapter 8 Solutions

Bundle: Calculus: Early Transcendental Functions, 7th + Webassign, Multi-term Printed Access Card

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The approximate value of definite integral ∫ 0 π / 4 x tan x d x for n = 4 by Trapezoidal rule and Simpson’s rule and compare it with the approximate value of the integral using graphing utility.

Solution Summary: The author compares the approximate value of the integral by Trapezoidal rule and Simpson's rule using graphing utility.

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To calculate: The approximate value of definite integral ∫0π/4xtanxdx for n=4 by Trapezoidal rule and Simpson’s rule and compare it with the approximate value of the integral using graphing utility.

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Chapter 8 Solutions