a. Explanation Given: The function y = cos − 1 x on the closed interval [ 0 , 1 ] . Definition used: “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 t h 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 . That is, 1. Left Riemann sum if x k * = a + ( k − 1 ) Δ x . 2. Right Riemann sum if x k * = a + k Δ x .” Calculation: The calculation of the left and right Riemann sums in sigma notation for an arbitrary value of n for the function f ( x ) = cos − 1 x on the interval [ 0 , 1 ] is as follows. Divide the interval [ 0 , 1 ] into n equal subintervals. Calculate the length of the each subinterval as follows. Δ x = b − a n = 1 − 0 n = 1 n Therefore, the length of each sub interval is, Δ x = 1 n . The interval [ 0 , 1 ] is divided into n equal subintervals with the grid points x 0 = 0 , x 1 = 1 n , x 2 = 2 n , x 3 = 3 n , ⋯ , x n = n n . Therefore, the sub intervals are [ 0 , 1 n ] , [ 1 n , 2 n ] , [ 2 n , 3 n ] , .......... , and [ ( n − 1 ) n , 1 ] . Thus the left end points of the intervals are 0 , 1 n , 1 n , 3 n , .......... , ( n − 1 ) n , and the right end points of the intervals are 1 n , 2 n , 3 n , .......... , 1 . Use the left Riemann sum formula and write the sigma notation of the left Riemann sum as follows. ∑ k = 1 n f ( x k * ) Δ x = f ( x 1 * ) Δ x + f ( x 2 * ) Δ x + f ( x 3 * ) Δ x + f ( x 4 * ) Δ x + .......... + f ( x n * ) Δ x = f ( 0 ) ( 1 n ) + f ( 1 n ) ( 1 n ) + f ( 2 n ) ( 1 n ) + f ( 3 n ) ( 1 n ) + .......... + f ( ( n − 1 ) n ) ( 1 n ) = [ cos − 1 ( 0 ) + cos − 1 ( 1 n ) + cos − 1 ( 2 n ) + cos − 1 ( 3 n ) + b. To determine To evaluate: The each sum using the calculator with n = 20 , 50 , and 100 , then use these values to estimate the value of the integral.

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 5.2, Problem 73E

To determine

To write: The left and right Riemann sums in sigma notation for an arbitrary value of n.

To determine

To evaluate: The each sum using the calculator with n=20, 50, and 100, then use these values to estimate the value of the integral.

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Chapter 5.2, Problem 72E

Chapter 5.2, Problem 74E

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The left and right Riemann sums in sigma notation for an arbitrary value of n .

Solution Summary: The author explains the sigma notation of left and right Riemann sums for an arbitrary value of n.