a. Explanation Definition used: Suppose f is defined and integrable on [ a , b ] and n ≥ 2 is an even integer. The Simpson Rule approximation to ∫ a b f ( x ) d x using n equally spaced subintervals on [ a , b ] is, S ( n ) = ( f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 + ... + 4 f ( x n − 1 ) + f ( x n ) ) ) Δ x 3 Where, Δ x = b − a n and x k = a + k Δ x , for k = 1 , 2 , 3... n . Calculation: The given integral is ∫ 1 2 sin x d x . For n = 30 . Compute the value of Δ x . Δ x = b − a n = 2 − 1 20 = 1 20 = 0.05 Thus, by above mentioned formula, x 0 = 1 + 0 ( 1 20 ) = 1 x 1 = 1 + 1 ( 1 20 ) = 21 20 The value of x 2 is, x 2 = 1 + 2 ( 1 20 ) = 11 10 Similarly, compute the value of x 3 , x 4 , x 5 ... x 20 by using the online Simpson rule calculator. Use Simpson formula and online Simpson calculator to compute the value of S ( 20 ) b. To determine The absolute error’s upper bound in the Simpson rule approximation obtained in part a .

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Author: Bill Briggs

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Chapter 8.8, Problem 71E

To determine

The approximation for the Simpson Rule to the integral ∫12sinxdx with subinterval n=20.

To determine

The absolute error’s upper bound in the Simpson rule approximation obtained in part a.

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Chapter 8.8, Problem 70E

Chapter 8.8, Problem 72E

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The approximation for the Simpson Rule to the integral ∫ 1 2 sin x d x with subinterval n = 20 .

Solution Summary: The author describes the Simpson Rule approximation to the integral displaystyle