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: Let us suppose that f ( x ) = x sin ( x ) . Since n = 4 Therefore, add π − π / 2 4 = π 8 to each term to get values of x 0 , x 2 ....... x m where x 0 = π / 2 and x 4 = π . Thus, the values of x are: x 1 = π / 2 + π / 8 = 5 π 8 x 2 = 5 π 8 + π 8 = 3 π 4 And, x 3 = 3 π 4 + π 8 = 7 π 8 x 4 = 7 π 8 + π 8 = π Now, by Trapezoidal Rule: Here, b = π , a = π 2 and n = 4 . Therefore, ∫ π / 2 π x sin ( x ) d x ≈ π − π / 2 2 ( 4 ) [ f ( π 2 ) + 2 f ( 5 π 8 ) + 2 f ( 3 π 4 ) + 2 f ( 7 π 8 ) + f ( π ) ] = π 16 [ π 2 sin ( π 2 ) + 2 5 π 8 s i n ( 5 π 8 ) + 2 3 π 4 sin ( 3 π 4 ) + 2 7 π 8 sin ( 7 π 8 ) + π sin ( π ) ] = π 16 [ π 2 + 2 5 π 8 ( 0

11th Edition

ISBN: 9781337275361

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differ…

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Chapter 8.6, Problem 22E

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To calculate: The approximate value of definite integral ∫π/2πxsin(x)dx 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

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The approximate value of definite integral ∫ π / 2 π x sin ( 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 Trapezoidal Rule and Simpson's Rule with the approximate value of the integral using graphing utility.

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To calculate: The approximate value of definite integral ∫π/2πxsin(x)dx for n=4 by Trapezoidal Rule and Simpson’s Rule and compare it with the approximate value of the integral using graphing utility.

Chapter 8 Solutions