Explanation Given information: The curve function is f ( x ) = 2 + sin x . The region lies between x = 0 and x = π . So the limits are a = 0 and b = π . Calculation: Draw the graph for the function f ( x ) = 2 + sin x with two rectangles as shown in Figure (1). Calculation: Find the width ( Δ x ) using the relation: Δ x = b − a n (1) Here, the upper limit is b , the lower limit is a , and the number of rectangles is n . The expression to find the upper sum of areas of n th rectangle is shown below. Upper sum = f ( x 1 ) Δ x + f ( x 2 ) Δ x + ... + f ( x n ) Δ x (2) Here, the maximum height of the first rectangle is f ( x 1 ) , the width is Δ x , the maximum height of the second rectangle is f ( x 2 ) , and the maximum height of n th rectangle is f ( x n ) Find the upper sum of areas for two rectangles as shown below. Substitute π for b , 0 for a , and 2 for n in Equation (1). Δ x = π − 0 2 = π 2 Refer to Figure (1). The maximum values of the function occur at x = π 2 on the both subintervals. Take the maximum height of the first rectangle’s f ( x 1 ) value as 3 and the maximum height of the second rectangle’s f ( x 2 ) value as 3. Substitute 2 for n , 3 for f ( x 1 ) , π 2 for Δ x , and 3 for f ( x 2 ) in Equation (2). Upper sum = ( 3 × π 2 ) + ( 3 × π 2 ) = 3 π + 3 π 2 = 6 π 2 = 9.425 Therefore, the upper sum of the function for two rectangles is 9.425 _ . The expression to find the lower sum of areas of n th rectangle is shown below. Lower sum = f ( x 0 ) Δ x + f ( x 1 ) Δ x + ... + f ( x n − 1 ) Δ x (3) Here, the minimum height of the first rectangle is f ( x 0 ) , the minimum height of the second rectangle is f ( x 1 ) , and the minimum height of n th rectangle is f ( x n − 1 ) Find the lower sum of areas for two rectangles as shown below. Refer to Figure (1). The minimum values of the function occur at x = 0 and x = π . Take the minimum height of the first rectangle’s f ( x 0 ) value as 2 and the minimum height of the second rectangle’s f ( x 1 ) value as 2. Substitute 2 for n , 2 for f ( x 0 ) , π 2 for Δ x , and 2 for f ( x 1 ) in Equation (3). Lower sum = ( 2 × π 2 ) + ( 2 × π 2 ) = 2 π + 2 π 2 = 4 π 2 = 6.283 Therefore, the lower sum of the function for two rectangles is 6.283 _ . Draw the graph for the function f ( x ) = 2 + sin x with four rectangles as shown in Figure (2). Find the upper sum of areas for four rectangles as shown below. Substitute π for b , 0 for a, and 4 for n in Equation (1). Δ x = π − 0 4 = π 4 Refer to Figure (2). Take the maximum height of the first rectangle’s f ( x 1 ) value as 2.7, the maximum height of the second rectangle’s f ( x 2 ) value as 3, the maximum height of the third rectangle’s f ( x 3 ) value as 3, and the maximum height of the fourth rectangle’s f ( x 4 ) value as 2.7. Substitute 4 for n , 2.7 for f ( x 1 ) , π 4 for Δ x , 3 for f ( x 2 ) , 3 for f ( x 3 ) , and 2.7 for f ( x 4 ) in Equation (2). Upper sum = ( 2.7 × π 4 ) + ( 3 × π 4 ) + ( 3 × π 4 ) + ( 2.7 × π 4 ) = 2.7 π + 3 π + 3 π + 2.7 π 4 = 11.4 π 4 = 8.954 Therefore, the upper sum of the function for four rectangles is 8.954 _ . Find the lower sum of areas for four rectangles as shown below. Refer to Figure (2). Take the minimum height of the first rectangle’s f ( x 0 ) value as 2, the minimum height of the second rectangle’s f ( x 1 ) value as 2.7, the minimum height of the third rectangle’s f ( x 2 ) value as 2.7, and the minimum height of the fourth rectangle’s f ( x 3 ) value as 2

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

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Chapter 5.1, Problem 7E

To determine

To evaluate:

The upper and lower sums for the function with n=2.

The upper and lower sums for the function with n=4.

The upper and lower sums for the function with n=8.

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Chapter 5.1, Problem 6E

Chapter 5.1, Problem 8E

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Chapter 5 Solutions

Single Variable Calculus: Early Transcendentals

Ch. 5.1 - Prob. 1E Ch. 5.1 - (a) Use six rectangles to find estimates of each...Ch. 5.1 - (a) Estimate the area under the graph of f(x) =...Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - The speed of a runner increased steadily during...Ch. 5.1 - The table shows speedometer readings at 10-second...

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The upper and lower sums for the function with n = 2 . The upper and lower sums for the function with n = 4 . The upper and lower sums for the function with n = 8 .

Solution Summary: The author evaluates the upper and lower sums for the function f(x)=2+mathrmsin

To evaluate:

The upper and lower sums for the function with n=2.

The upper and lower sums for the function with n=4.

The upper and lower sums for the function with n=8.

Want to see the full answer?

Chapter 5 Solutions

The upper and lower sums for the function with n = 2 . The upper and lower sums for the function with n = 4 . The upper and lower sums for the function with n = 8 .

Solution Summary: The author evaluates the upper and lower sums for the function f(x)=2+mathrmsin

To evaluate: The upper and lower sums for the function with n=2. The upper and lower sums for the function with n=4. The upper and lower sums for the function with n=8.

Want to see the full answer?

Chapter 5 Solutions

To evaluate:

The upper and lower sums for the function with n=2.

The upper and lower sums for the function with n=4.

The upper and lower sums for the function with n=8.