Explanation Given: The exact value of the function is f ( x ) = ∫ 0 6 3 e − 3 x d x = 1.000000 . Definition used: Suppose f is defined and integrable on [ a , b ] . The Trapezoid Rule approximation to ∫ a b f ( x ) d x that uses n equally spaced subintervals on [ a , b ] is, T ( n ) = ( 1 2 f ( x 0 ) + ∑ k = 1 n − 1 f ( x k ) + 1 2 f ( x n ) ) Δ x , where Δ x = b − a n and x 0 = a , x k = a + k Δ x , for k = 1 , 2 , 3... n . Calculation: Consider the case n = 4 . Compute the value of Δ x . Δ x = b − a n = 6 − 0 4 = 6 4 = 1.5 Thus, by above mentioned formula, x 0 = 0 , x 1 = 1.5 …. x 4 = 6 . Compute the value of T ( 4 ) and use online calculator. T ( 4 ) = T ( n ) = ( 1 2 f ( x 0 ) + ∑ k = 1 1 f ( x k ) + 1 2 f ( x 4 ) ) Δ x = ( 1 2 f ( 0 ) + f ( 1.5 ) + ⋯ + 1 2 f ( 6 ) ) ( 1.5 ) = ( 1 2 ∫ 0 6 3 e − 3 ( 0 ) d x + ∫ 0 6 3 e − 3 ( 1.5 ) d x + ⋯ + 1 2 ∫ 0 6 3 e − 3 ( 6 ) d x ) ⋅ 1.5 = 2.30055 Now consider the case n = 8 . Compute the value of Δ x . Δ x = b − a n = 6 − 0 8 = 6 8 = 0.75 Thus, by above mentioned formula, x 0 = 0 , x 1 = 0.75 …. x 8 = 6 . Compute the value of T ( 8 ) and use online calculator. T ( 8 ) = T ( n ) = ( 1 2 f ( x 0 ) + ∑ k = 1 1 f ( x k ) + 1 2 f ( x 8 ) ) Δ x = ( 1 2 f ( 0 ) + f ( 0.75 ) + ⋯ + 1 2 f ( 6 ) ) ( 0.75 ) = ( 1 2 ∫ 0 6 3 e − 3 ( 0 ) d x + ∫ 0 6 3 e − 3 ( 0.75 ) d x + ⋯ + 1 2 ∫ 0 6 3 e − 3 ( 6 ) d x ) ⋅ 0.75 = 1.39009 Consider the case n = 16 . Compute the value of Δ x . Δ x = b − a n = 6 − 0 16 = 6 16 = 0.375 Thus, by above mentioned formula, x 0 = 0 , x 1 = 0.375 …. x 16 = 6 . Compute the value of T ( 16 ) and use online calculator. T ( 16 ) = T ( n ) = ( 1 2 f ( x 0 ) + ∑ k = 1 1 f ( x k ) + 1 2 f ( x 16 ) ) Δ x = ( 1 2 f ( 0 ) + f ( 0.375 ) + ⋯ + 1 2 f ( 6 ) ) ( 0.375 ) = ( 1 2 ∫ 0 6 3 e − 3 ( 0 ) d x + ∫ 0 6 3 e − 3 ( 0.375 ) d x + ⋯ + 1 2 ∫ 0 6 3 e − 3 ( 6 ) d x ) ⋅ 0

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Chapter 8.8, Problem 52E

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To find: The value of the integral ∫063e−3xdx, the errors when n=4,8,16 and 32, and then construct a table of values by using Simpson’s rule approximation.

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Chapter 8.8, Problem 51E

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The value of the integral ∫ 0 6 3 e − 3 x d x , the errors when n = 4 , 8 , 16 and 32, and then construct a table of values by using Simpson’s rule approximation.

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To find: The value of the integral ∫063e−3xdx, the errors when n=4,8,16 and 32, and then construct a table of values by using Simpson’s rule approximation.

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