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 ) / 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 ( 3 − 2 x ) d x . So, the function is f ( x ) = 3 − 2 x . The given interval is [ a , b ] = [ − 1 , 5 ] and 2 n = 6 . Divide the interval [ − 1 , 5 ] into six equal subintervals. So, the length of each subinterval becomes, Δ x = ( b − a ) / 2 n = ( 5 − ( − 1 ) ) / 6 = 6 / 6 = 1 Thus, the length of each interval is 1. So, the end points of the each subinterval are x 0 = − 1 , x 1 = 0 , x 2 = 1 , x 3 = 2 , x 4 = 3 , x 5 = 4 and x 6 = 5 . Calculate the value of the function f ( x ) = ( 3 − 2 x ) at each end point as follows. x f ( x ) − 1 ( 3 − 2 x ) = ( 3 − 2 ( − 1 ) ) = 5 0 ( 3 − 2 x ) = ( 3 − 2 ( 0 ) ) = 3 1 ( 3 − 2 x ) = ( 3 − 2 ( 1 ) ) = 1 2 ( 3 − 2 x ) = ( 3 − 2 ( 2 ) ) = − 1 3 ( 3 − 2 x ) = ( 3 − 2 ( 3 ) ) = − 3 4 ( 3 − 2 x ) = ( 3 − 2 ( 4 ) ) = − 5 5 ( 3 − 2 x ) = ( 3 − 2 ( 5 ) ) = − 7 Table 1 Calculate S 6 by using Simpson’s rule

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

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Chapter 6.4, Problem 36E

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To show: The approximate value of the integral ∫−15(3−2x)dx using the Simpson’s rule is same as the exact value of ∫−15(3−2x)dx using fundamental theorem of calculus.

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Chapter 6.4, Problem 35E

Chapter 6.4, Problem 37E

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The approximate value of the integral ∫ − 1 5 ( 3 − 2 x ) d x using the Simpson’s rule is same as the exact value of ∫ − 1 5 ( 3 − 2 x ) d x using fundamental theorem of calculus.

Solution Summary: The author illustrates the Simpson's rule by comparing the approximate value of the integral with the exact value using fundamental theorem of calculus.

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To show: The approximate value of the integral ∫−15(3−2x)dx using the Simpson’s rule is same as the exact value of ∫−15(3−2x)dx using fundamental theorem of calculus.

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