Explanation Given: y = 1 4 x 4 Over the interval [ 1 , 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 Trapezoidal Rule: ∫ a b f ( x ) d x ≈ T N = Δ x 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + 2 f ( x 3 ) + ........... + 2 f ( x n − 1 ) + f ( x n ) ] Where Δ x = b − a n d d x x n = n x n − 1 Calculation: To calculate the arc length we first differentiate the given function and then use the formula to find the arc length and Trapezoidal rule. y = 1 4 x 4 g ' ( x ) = 1 4 d d x ( x 4 ) = 4 4 x 4 − 1 g ' ( x ) = x 3 1 + ( g ' ( x ) ) 2 = 1 + ( x 3 ) 2 = 1 + x 6 Arc length= s = ∫ 1 2 1 + x 6 d x where a = 1 & b = 2 Now, using Trapezoidal rule ∫ a b f ( x ) d x ≈ T N = Δ x 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + 2 f ( x 3 ) + ........... + 2 f ( x n − 1 ) + f ( x n ) ] For s = ∫ 1 2 1 + x 6 d x , f ( x ) = 1 + x 6 and Δ x = b − a n = 2 − 1 5 = 1 5 = 0...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 9.2, Problem 11E

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the arc length using Trapezoidal Rule $T_{5}$

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the arc length using Trapezoidal Rule T 5

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the arc length using Trapezoidal Rule T 5

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the arc length using Trapezoidal Rule $T_{5}$