To determine To approximate: The definite integral ∫ 2 5 ( 0.25 x 2 − 4 ) d x using left and right sum with n = 6 and compute its error bounds as well as describe its geometrical representation over the interval [ 2 , 5 ] . Explanation Given: The interval is [ 2 , 5 ] and the corresponding interval is divided into 6 subintervals. The definite integral is ∫ 2 5 ( 0.25 x 2 − 4 ) d x . Formula used: “Let f be a function defined on the interval [ a , b ] and partition the interval into n subintervals of equal length Δ x = b − a n with endpoints a = x 0 < x 1 < x 2 < … < x n = b . Then, Left sum is defined: L n = f ( x 1 ) Δ x + f ( x 2 ) Δ x + … + f ( x n − 1 ) Δ x . Theorem used: ( Error bounds for approximation of area by left or right sums ) “Let f ( x ) > 0 and is either increasing or decreasing on [ a , b ] , then | f ( b ) − f ( a ) | b − a n is an error bound for the approximation of the area between the graph of f and the x axis, from x = a to x = b , by L n or R n .” Calculation: From the definite integral the integrand is y = 0.25 x 2 − 4 . Divide the interval [ 2 , 5 ] into 6 subinterval. Then, the equal length Δ x = 5 − 2 6 = 3 6 = 0.5 By the help of the equal length, the endpoints becomes, a = 2 < 2.5 < 3.5 < 4.5 < 5 = b The left end points are 2, 2.5, 3, 3.5,4 and 4.5, respectively. The right end points are 2.5, 3, 3.5,4,4.5 and 5, respectively. Calculate the value of y = 0.25 x 2 − 4 at x 1 = 2 . f ( 2 ) = 0.25 ( 2 ) 2 − 4 = 1 − 4 = − 3 Calculate the value of y = 0.25 x 2 − 4 at x 2 = 2.5 . f ( 2.5 ) = 0.25 ( 2.5 ) 2 − 4 = 1.5625 − 4 = − 2.4375 Calculate the value of y = 0.25 x 2 − 4 at x 3 = 3 . f ( 3 ) = 0.25 ( 3 ) 2 − 4 = 2.25 − 4 = − 1.75 Calculate the value of y = 0.25 x 2 − 4 at x 4 = 3.5 . f ( 3.5 ) = 0.25 ( 3.5 ) 2 − 4 = 3.0625 − 4 = − 0.9375 Calculate the value of y = 0.25 x 2 − 4 at x 4 = 4

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

14th Edition

ISBN: 9780134674148

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

Publisher: PEARSON

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Chapter 12.4, Problem 63E

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To approximate: The definite integral ∫25(0.25x2−4)dx using left and right sum with n=6 and compute its error bounds as well as describe its geometrical representation over the interval [2,5].

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Chapter 12.4, Problem 62E

Chapter 12.4, Problem 64E

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

Ch. 12.1 - Matched Problem 1 Note that Find all...Ch. 12.1 - Prob. 2MP Ch. 12.1 - Prob. 3MP Ch. 12.1 - Prob. 4MP Ch. 12.1 - Prob. 5MP Ch. 12.1 - Prob. 6MP Ch. 12.1 - Prob. 1ED Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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