Explanation Given Information: The definite integral is ∫ 1 2 1 x 3 d x , n = 8 . Formula used: According to Trapezoidal Rule if a function f is continuous on [ a , b ] , then, ∫ a b f ( x ) d x = ( b − a 2 n ) [ f ( x 0 ) + 2 f ( x 1 ) + ⋯ + 2 f ( x n − 1 ) + f ( x n ) ] According to Simpson’s Rule if f is continuous on [ a , b ] and n is an even integer, then, ∫ a b f ( x ) d x = ( b − a 3 n ) [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + ⋯ + 4 f ( x n − 1 ) + f ( x n ) ] . Calculation: Calculation to get exact value: Consider the definite integral ∫ 1 2 1 x 3 d x . ∫ 1 2 1 x 3 d x = [ − 1 2 x 2 ] 1 2 = [ − 1 2 ( 2 ) 2 − ( − 1 2 ( 1 ) 2 ) ] = − 1 + 4 8 = 3 8 Simplify as: ∫ 1 2 1 x 3 d x = 3 8 = 0.3750 Therefore, the exact value of the integral is 0.375. Calculation by Trapezoidal Rule: Consider the definite integral ∫ 1 2 1 x 3 d x . When n = 8 , the width of each subinterval is: 2 − 1 8 = 1 8 And the end points of subintervals are x 0 = 1 , x 1 = 1 + 1 8 = 9 8 , x 2 = 9 8 + 1 8 = 10 8 = 5 4 , x 3 = 5 4 + 1 8 = 10 + 1 8 = 11 8 , x 4 = 11 8 + 1 8 = 12 8 = 6 4 = 3 2 , x 5 = 3 2 + 1 8 = 12 + 1 8 = 13 8 , x 6 = 13 8 + 1 8 = 14 8 = 7 4 , x 7 = 7 4 + 1 8 = 14 + 1 8 = 15 8 , x 8 = 15 8 + 1 8 = 16 8 = 2 By using Trapezoidal Rule: ∫ 1 2 1 x 3 d x = 2 − 1 2 × 8 [ 1 1 2 + 2 ( 1 ( 9 8 ) 3 ) + 2 ( 1 ( 5 4 ) 3 ) + 2 ( 1 ( 11 8 ) 3 ) + 2 ( 1 ( 3 2 )

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Author: Larson

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Chapter 6, Problem 36RE

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To calculate: The approximate value of the integral ∫121x3dx with n=8 by using the Trapezoidal Rule and Simpson’s Rule and compare the results with the exact value of definite integral.

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Chapter 6, Problem 35RE

Chapter 6, Problem 37RE

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Chapter 6 Solutions

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The approximate value of the integral ∫ 1 2 1 x 3 d x with n = 8 by using the Trapezoidal Rule and Simpson’s Rule and compare the results with the exact value of definite integral.

Solution Summary: The author calculates the approximate value of the integral using the Trapezoidal Rule and Simpson’s Rule.

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To calculate: The approximate value of the integral ∫121x3dx with n=8 by using the Trapezoidal Rule and Simpson’s Rule and compare the results with the exact value of definite integral.

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Chapter 6 Solutions