Explanation Find the value of Δ x and using the formula of R N = Δ x ∑ j = 1 N f ( x j ) and S N = Δ x ∑ j = 1 N f ( x j ) to determine the value of ∫ 0 b x k d x Given: k and b are positive Calculation: Since k and b are positive numbers and assuming f ( x ) = x k on [ 0 , b ] . Δ x = ( b − 0 ) / N = b / N where N is a positive integer. The right-end points of the N subintervals of [ 0 , b ] can be given by x j = a + j Δ x = b j / N , where j = 1 , 2 , 3 , ... , N . Therefore the right-end point approximation to ∫ 0 b f ( x ) d x = ∫ 0 b x k d x is given by R N = Δ x ∑ j = 1 N f ( x j ) = b N ∑ j = 1 N ( b j N ) k = b

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 5.2, Problem 85E

To determine

To prove that $\int_{0}^{b} x^{k} d x = b^{k + 1} \int_{0}^{1} x^{k} d x$ by comparing the right-end point approximations

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Chapter 5.2, Problem 84E

Chapter 5.2, Problem 86E

Chapter 5 Solutions

CALCULUS (CLOTH)

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To prove that ∫ 0 b x k d x = b k + 1 ∫ 0 1 x k d x by comparing the right-end point approximations

Solution Summary: The author explains how the right-end point approximation to 0 b x k d is given by the formula of R N and S N.

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To prove that $\int_{0}^{b} x^{k} d x = b^{k + 1} \int_{0}^{1} x^{k} d x$ by comparing the right-end point approximations

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To prove that ∫ 0 b x k d x = b k + 1 ∫ 0 1 x k d x by comparing the right-end point approximations

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To prove that $\int_{0}^{b} x^{k} d x = b^{k + 1} \int_{0}^{1} x^{k} d x$ by comparing the right-end point approximations