Explanation Given: The integral function as ∫ 0 π sin x d x Number of rectangles n = 5 , 10 , 50 and 100 . Calculation: Show the Equation of the integral as follows: ∫ 0 π sin x d x (1) Consider the value of the function f ( x ) = sin x (2) For n = 5 Find the width ( Δ x ) using the relation: Δ x = b − a n (3) Here, the upper limit is b , the lower limit is a , and the number of rectangles is n . The limits of the integral ∫ 0 π sin x d x , a = 0 and b = π . Substitute π for b , 5 for n, and 0 for a in Equation (3). Δ x = π − 0 5 = π 5 The right endpoints are x 1 = π 5 , x 2 = 2 π 5 , x 3 = 3 π 5 , x 4 = 4 π 5 and x 5 = π . The expression to find the right Riemann sum R n as shown below: R n = ∑ i = 1 n f ( x i ) Δ x = f ( x 1 ) Δ x + f ( x 2 ) Δ x + ... + f ( x n ) Δ x (4) Here, the right endpoint height of first rectangle is f ( x 1 ) , the width is Δ x , height of right endpoint of second rectangle is f ( x 2 ) , and left endpoint height of n th rectangle is f ( x n ) . Calculate the value of f ( x 1 ) using the Equation (2). Substitute π 5 for x 1 in Equation (2). f ( x 1 ) = sin x 1 = sin ( π 5 ) = 0.0109 Calculate the value of f ( x 2 ) using the Equation (2). Substitute 2 π 5 for x 2 in Equation (2). f ( x 2 ) = sin x 2 = sin ( 2 π 5 ) = 0.02193 Similarly calculate the value of f ( x ) for remaining values of x . Calculate the right Riemann sum using calculator. Substitute π 5 for Δ x , 5 for n, and values of right end points in Equation (4). R n = ∑ i = 1 5 f ( x i ) Δ x = f ( x 1 ) Δ x + f ( x 2 ) Δ x + ... + f ( x 5 ) Δ x = f ( π 5 ) × π 5 + f ( 2 π 5 ) × π 5 + ... + f ( π ) × π 5 = 1.933766 The value of right Riemann Sum for n = 5 is 1.933766 _ . For n = 10 Find the width ( Δ x ) using the Equation (3): Substitute π for b , 10 for n, and 0 for a in Equation (3). Δ x = π − 0 10 = π 10 The right endpoints are x 1 = π 10 , x 2 = 2 π 10 , x 3 = 3 π 10 x 10 = π . Calculate the value of f ( x 1 ) using the Equation (2). Substitute π 10 for x 1 in Equation (2). f ( x 1 ) = sin x 1 = sin ( π 10 ) = 5.483 × 10 − 3 Calculate the value of f ( x 2 ) using the Equation (2). Substitute 2 π 10 for x 2 in Equation (2). f ( x 2 ) = sin x 2 = sin ( 2 π 10 ) = 0.01096 Similarly calculate the value of f ( x ) for remaining values of x . Calculate the right Riemann sum using calculator. Substitute π 10 for Δ x , 10 for n, and values of right end points in Equation (4). R 10 = ∑ i = 1 10 f ( x i ) Δ x = f ( x 1 ) Δ x + f ( x 2 ) Δ x + ... + f ( x 10 ) Δ x = f ( π 10 ) × π 10 + f ( 2 π 10 ) × π 10 + ... + f ( π ) × π 10 = 1

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 4.2, Problem 15E

To determine

To Calculate: The right Riemann sums Rn for the integral ∫0πsinx dx for n=5,10,50, and100 using the calculator.

To Find: The values of the numbers appear to be approaching.

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Chapter 4.2, Problem 14E

Chapter 4.2, Problem 16E

Chapter 4 Solutions

Single Variable Calculus

Ch. 4.1 - Prob. 1E Ch. 4.1 - (a) Use six rectangles to find estimates of each...Ch. 4.1 - (a) Estimate the area under the graph of f(x) =...Ch. 4.1 - Prob. 4E Ch. 4.1 - (a) Estimate the area under the graph of f(x) = 1...Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - With a programmable calculator (or a computer), it...Ch. 4.1 - With a programmable calculator (or a computer), it...

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