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 + .... + y N − 1 ) + 2 ( y 2 + y 4 + ..... y N − 2 ) ] (4) Given: From (2) and (3), a = x 0 = 0 , b = x 10 = 2 , h = b − a N = 2 10 = 0...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 8.8, Problem 24E

To determine

the approximate value $S_{10}$ of, $\int_{0}^{2} {(x^{2} + 1)}^{\frac{- 1}{3}} d x$ using Simpson’s rule, with $N = 10$

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Chapter 8.8, Problem 23E

Chapter 8.8, Problem 25E

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approximate the following two areas with the given specifications. If your answer involves a decimal, round your final answer to the nearest 0.001 a) The area bounded byf(x)=8x-x^2 and the x-axis between x=1 and x=4. Use n=6. Determine the LHA, RHA, and the average of the two. b) The area bounded by f(x) =e^x and the x-axis between x=0 and x=2. Use n=6. Determine the LHA, RHA, and the average of the two.

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Chapter 8 Solutions

CALCULUS (CLOTH)

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