Explanation Definition used 1: “Let f be the function defined on a closed interval [ a , b ] , which is divided into n subintervals of equal length Δ x . If x k * is a point in the k th subinterval [ x k − 1 , x k ] , for k = 1 , 2 , .... n , then the Riemann sum for f on [ a , b ] is ∑ k = 1 n f ( x k * ) Δ x . Definition used 2: The limit definition of the definite integral is given by ∫ a b f ( x ) d x = lim n → ∞ ∑ k = 1 n f ( x k * ) Δ x . Theorem: Fundamental theorem of calculus : “If f is continuous on [ a , b ] and F is any anti-derivative of f on [ a , b ] , then ∫ a b f ( x ) d x = F ( b ) − F ( a ) ”. Given: The definite integral ∫ 0 2 ( x 2 − 4 ) d x . Calculation: According to the given data, a = 0 and b = 2 . Divide the interval [ 0 , 2 ] into n equal subintervals. Calculate the length of the each subinterval as follows. Δ x = b − a n = 2 − 0 n = 2 n Therefore, the length of each sub interval is, Δ x = 2 n . By the definition, Right Riemann sum if x k * = a + k Δ x . That is, x k * = 0 + k 2 n x k * = 2 k n Consider ∑ k = 1 n f ( x k * ) Δ x (1) By the above information, the equation (1) can be calculated as follows. ∑ k = 1 n f ( x k * ) Δ x = ∑ k = 1 n ( ( 2 k n ) 2 − 4 ) 2 n = ∑ k = 1 n ( 4 k 2 n 2 − 4 ) 2 n = ∑ k = 1 n ( 8 k 2 n 3 − 8 n ) = 8 n 3 ∑ k = 1 n k 2 − 8 n ∑ k = 1 n 1

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 5, Problem 11RE

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To evaluate: The value of the right Riemann sum by using the limit definition of the definite integral and check the required result by using Fundamental theorem of calculus.

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Chapter 5, Problem 10RE

Chapter 5, Problem 12RE

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Calculus, Single Variable: Early Transcendentals (3rd Edition)

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The value of the right Riemann sum by using the limit definition of the definite integral and check the required result by using Fundamental theorem of calculus.

Solution Summary: The author evaluates the right Riemann sum by using the limit definition of the definite integral and checks the required result using Fundamental theorem of calculus.

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To evaluate: The value of the right Riemann sum by using the limit definition of the definite integral and check the required result by using Fundamental theorem of calculus.

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Chapter 5 Solutions