Explanation Given: The provided integral is: ∫ 0 1 x 1 − x d x 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 1 − x Since, n = 4 Therefore, add 1 − 0 4 = 1 4 to each term to get values of x 0 , x 2 ....... x m where x 0 = 0 and x 4 = 1 . Thus, the values of x are: x 1 = 0 + 1 4 = 1 4 x 2 = 1 4 + 1 4 = 1 2 And x 3 = 1 2 + 1 4 = 3 4 x 4 = 3 4 + 1 4 = 1 Now, by Trapezoidal rule: Here, b = 1 , a = 0 and n = 4 . Therefore, ∫ 0 1 x 1 − x d x ≈ 1 − 0 2 ( 4 ) [ f ( 0 ) + 2 f ( 1 4 ) + 2 f ( 1 2 ) + 2 f ( 3 4 ) + f ( 1 ) ] = 1 8 [ 0 1 − 0 + 2 ( 1 / 4 1 − 1 4 ) + 2 ( 1 / 2 1 − 1 2 ) + 2 ( 3 / 4 1 − 3 4 ) + ( 1 1 − 1 ) ]

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 16E

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To calculate: The approximate value of definite integral ∫01x1−xdx 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 15E

Chapter 8.6, Problem 17E

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The approximate value of definite integral ∫ 0 1 x 1 − 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 ∫01x1−xdx for n=4 by Trapezoidal rule and Simpson’s rule and compare it with the approximate value of the integral using graphing utility.

Want to see the full answer?

Chapter 8 Solutions