Explanation Given information: The curve function is y = sin x (1) The lower limit is a = 0 and the upper limit is b = 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 . The expression to find the area of the surface ( S ) obtained by rotating the curve about the x -axis is shown below: S = ∫ a b 2 π y 1 + ( d y d x ) 2 d x (4) Here, the derivative of the function y is d y d x , the lower limit is a , and the upper limit is b . Here, the derivative of the function y is d y d x . Differentiate Equation (1) with respect to x . d y d x = cos x Substitute sin x for y and cos x for d y d x in Equation (4). S = ∫ a b 2 π sin x 1 + cos 2 x d x (5) Let f ( x ) = 2 π sin x 1 + cos 2 x (6) Find the width of the interval using Equation (3): Substitute π for b , 0 for a , and 10 for n in Equation (3). Δ x = π − 0 10 = π 10 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 ) ] (7) Split the interval 0 to π into 10 subintervals. Thus the intervals are 0 , π 10 , 2 π 10 , 3 π 10 , 4 π 10 , 5 π 10 , 6 π 10 , 7 π 10 , 8 π 10 , 9 π 10 , and π

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Author: James Stewart

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Chapter 8, Problem 8RE

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To find: The area of the surface obtained by rotating the sine curve about x-axis using Simpson’s rule.

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The area of the surface obtained by rotating the sine curve about x -axis using Simpson’s rule.

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