Exact Simpson’s Rule Prove that Simpson’s Rule is exact

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Problem 57AE Chapter 7.6

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 57AE

Exact Simpson’s Rule

Prove that Simpson’s Rule is exact (no error) when approximating the definite integral of a linear function and a quadratic function.

Step-by-Step Solution:
Step 1 of 3

Solution:-

Step1

Simpson’s rule

Three points integration rule derived using the method of undetermined coefficients.

Suppose that we add a quadrature point at the middle of the interval [a, b],

To avoid algebra,

substitute x =  and define h =

so that the integral becomes

 

Since we have three unknowns,

we can make this formula exact for all quadratic functions;

so, let us use,

 e.g., F ≡ 1, F ≡ u and F ≡ u 2 .

 F ≡ 1 ⇒  = 2h = w1 + w2 + w3;

 F ≡ u ⇒   = 0 = −hw1 + hw3 ⇒ w1 = w3;

F ≡  ⇒   =  = w1 + w3 ⇒ w1 = w3 =  ; w2 = 2h − w1 − w3 = 2h − h =

Hence we obtain the approximation

 

which...

Step 2 of 3

Chapter 7.6, Problem 57AE is Solved
Step 3 of 3

Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

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Exact Simpson’s Rule Prove that Simpson’s Rule is exact

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