Explanation Simpson’s rule is an estimate for the value of the definite integral ∫ a = x 0 b = x N f ( x ) d x , which is the area under the curve y = f ( x ) , between a = x 0 a n d b = x N ,N denoting the number of intervals into which the interval [a, b] is divided. The area is approximated as the sum of the areas under parabolas , resulting in a finite sum using the function values of f(x) at the division points. The case for N = 6 is illustrated in the adjoining figure: Formula used: Let N (an even positive integer) be given. Let the interval [a, b] be equally divided into N subintervals at the points x 0 = a , x 1 = a + h , x 2 = a + 2 h , ..... , x N − 1 = a + ( N − 1 ) h , x N = a + N h = b (1) With h = b − a N (2) Let y 0 = f ( x 0 ) = f ( a ) , y 1 = f ( x 1 ) , .... y i = f ( x i ) , ... y N = f ( x N ) = f ( b ) (3) (In other words , y i is the function value at the i th division point x i . Simpson’s formula for S N is given by S N = h 3 [ ( y 0 + y N ) + 4 ( y 1 + y 3 + ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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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 8.8, Problem 18E

To determine

The approximate value $S_{6}$ of $\int_{0}^{1} \cos (x^{2}) d x$ , using Simpson’s rule, with $N = 6$

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Chapter 8.8, Problem 17E

Chapter 8.8, Problem 19E

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00 (a) Starting with the geometric series Σ X^, find the sum of the series n = 0 00 Σηχη - 1, |x| < 1. n = 1 (b) Find the sum of each of the following series. 00 Σnx", n = 1 |x| < 1 (ii) n = 1 sin (c) Find the sum of each of the following series. (i) 00 Σn(n-1)x^, |x| <1 n = 2 (ii) 00 n = 2 n² - n 4n (iii) M8 n = 1 շո

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The approximate value $S_{6}$ of $\int_{0}^{1} \cos (x^{2}) d x$ , using Simpson’s rule, with $N = 6$