Explanation Given: y = e − x 2 Over the interval [ 0 , 2 ] Key Concept and Formula Used: The arc length s of function y = g ( x ) over [ a , b ] is equal to s = ∫ a b 1 + ( g ' ( x ) ) 2 d x Simpson’s Rule: ∫ a b f ( x ) d x ≈ S N = Δ x 3 [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + ........... + 4 f ( x n − 1 ) + f ( x n ) ] Where Δ x = b − a n d d x x n = n x n − 1 , d d x e x = e x Calculation: To calculate the arc length we first differentiate the given function and then use the formula to find the arc length and Simpson’s rule. y = e − x 2 g ' ( x ) = d d x ( e − x 2 ) = − 2 x e − x 2 1 + ( g ' ( x ) ) 2 = 1 + [ − 2 x e − x 2 ] 2 = 1 + 4 x 2 e − 2 x 2 Arc length= s = ∫ 0 2 1 + 4 x 2 e − 2 x 2 d x where a = 0 & b = 2 Now, using Simpson’s rule ∫ a b f ( x ) d x ≈ S N = Δ x 3 [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + ........... + 4 f ( x n − 1 ) + f ( x n ) ] For s = ∫ 0 2 1 + 4 x 2 e − 2 x 2 d x , f ( x ) = 1 + 4 x 2 e − 2 x 2 and Δ x = b − a n = 2 − 0 8 = 2 8 = 1 4 = 0.25 , so ∫ a b f ( x ) d x ≈ S N = Δ x 3 [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + 2 f ( x 4 ) + 4 f ( x 5 ) + 2 f ( x 6 ) + 4 f ( x 7 ) + f ( x 8 ) ] x 0 = 0 , x 1 = 0 + ...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 9.2, Problem 14E

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the arc length using Simpson’s Rule $S_{8}$

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the arc length using Simpson’s Rule S 8

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the arc length using Simpson’s Rule $S_{8}$

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the arc length using Simpson’s Rule S 8

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the arc length using Simpson’s Rule $S_{8}$