Explanation Given information: The curve function is y = x sin x (1) The lower limit is a = 0 and the upper limit is b = 2 π . The number of intervals is 10. Calculation: The Simpson’s rule is shown below: ∫ 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 n − 2 ) + 4 f ( x n − 1 ) + f ( x n ) ] (2) The interval width Δ x = b − a n (3) Here, the value of the function f ( x ) at x = 0 to n is f ( x 0 ) to f ( x n ) , the upper limit is b , the lower limit is a , and the number of interval is n . Find the value of f ( x ) using the relation: f ( x ) = 1 + ( d y d x ) 2 (4) Here, the derivative of the function y is d y d x . Differentiate Equation (1) with respect to x . d y d x = x cos x + sin x Substitute ( x cos x + sin x ) for d y d x in Equation (4). f ( x ) = 1 + ( x cos x + sin x ) 2 (5) Find the width of the interval using Equation (3): Substitute 2 π for b , 0 for a , and 10 for n in Equation (3). Δ x = 2 π − 0 10 = π 5 Modify Simpson’s rule for n = 10 . S 10 = Δ x 3 [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + ... + 2 f ( x 8 ) + 4 f ( x 9 ) + f ( x 10 ) ] (6) Split the interval 0 to 2 π into 10 subintervals. Thus the intervals are 0 , π 5 , 2 π 5 , 3 π 5 , 4 π 5 , 5 π 5 , 6 π 5 , 7 π 5 , 8 π 5 , 9 π 5 , and 2 π . Substitute 0 for x 0 , π 5 for x 1 , 2 π 5 for x 2 , 3 π 5 for x 3 , 4 π 5 for x 4 , π for x 5 , 6 π 5 for x 6 , 7 π 5 for x 7 , 8 π 5 for x 8 , 9 π 5 for x 9 , and 2 π for x 10 in Equation (6)

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ISBN: 9781133490975

Author: Stewart, James

Publisher: Cengage Learning

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Chapter 7.4, Problem 21E

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To find: The arc length of the curve.

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The arc length of the curve.

Solution Summary: The author calculates the arc length of the curve using Simpson's rule and calculator.

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