Explanation Formula used: Simpson’s rule: Let f be a function on an interval [ a , b ] . Partition [ a , b ] into 2 n subintervals of equal length Δ x = ( b − a ) / 2 n with end points a = x 0 < x 1 < x 2 < ... < x n = b . Then S 2 n = [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + 2 f ( x 4 ) .... + 4 f ( x 2 n − 1 ) + f ( x 2 n ) ] Δ x / 3 is an approximation of ∫ a b f ( x ) d x . Calculation: The given integral is ∫ 1 5 1 x d x . So, the function is f ( x ) = 1 x . The given interval is [ a , b ] = [ 1 , 5 ] and 2 n = 8 . Partition the interval [ 1 , 5 ] into 8 equal subintervals. The length of the each interval is Δ x = ( b − a ) / 2 n Substitute a = 1 , b = 5 and 2 n = 8 . Δ x = ( b − a ) 2 n = 5 − 1 8 = 4 8 = 0.5 Thus, the length of each interval is 0.5. So, the end points of the each subinterval are x 0 = 1 , x 1 = 1.5 , x 2 = 2 , x 3 = 2.5 , x 4 = 3 , x 5 = 3.5 , x 6 = 4 , x 7 = 4.5 and x 8 = 5 Calculate the value of the function f ( x ) = 1 x at each end point as follows. x f ( x ) 1 1 x = 1 1 = 1 1.5 1 x = 1 1.5 = 0.67 2 1 x = 1 2 = 0.5 2.5 1 x = 1 2.5 = 0.4 3 1 x = 1 3 = 0.333 3.5 1 x = 1 3.5 = 0.29 4 1 x = 1 4 = 0.25 4.5 1 x = 1 4.5 = 0.222 5 1 x = 1 5 = 0.2 Table 1 By Simpson’s rule, the formula is given by S 2 n = [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + 2 f ( x 4 )

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 6.4, Problem 31E

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The approximate value of the integral ∫151xdx using the Simpson’s rule and also to find the exact value of the integral ∫151xdx using fundamental theorem of calculus.

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Chapter 6.4, Problem 30E

Chapter 6.4, Problem 32E

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Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

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The approximate value of the integral ∫ 1 5 1 x d x using the Simpson’s rule and also to find the exact value of the integral ∫ 1 5 1 x d x using fundamental theorem of calculus.

Solution Summary: The author calculates the approximate value of the integral using the Simpson's rule and the fundamental theorem of calculus.

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The approximate value of the integral ∫151xdx using the Simpson’s rule and also to find the exact value of the integral ∫151xdx using fundamental theorem of calculus.

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Chapter 6 Solutions