Explanation Given information: The curve function is f ( x ) = 1 + x 2 . The region lies between x = − 1 and x = 1 . So the limits are a = − 1 and b = 1 . Calculation: Draw the graph for the function f ( x ) = 1 + x 2 with three rectangles as shown in Figure (1). 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 three rectangles as shown below. Substitute 1 for b , − 1 for a, and 3 for n in Equation (1). Δ x = 1 + 1 3 = 2 3 Refer to Figure (1). Take the maximum height of the first rectangle’s f ( x 1 ) value as 2, the maximum height of the second rectangle’s f ( x 2 ) value as 1.1, and the maximum height of the third rectangle’s f ( x 3 ) value as 2. Substitute 3 for n , 2 for f ( x 1 ) , 2 3 for Δ x , 1.1 for f ( x 2 ) , and 2 for f ( x 3 ) in Equation (2). upper sum = ( 2 × 2 3 ) + ( 1.1 × 2 3 ) + ( 2 × 2 3 ) = 4 + 2.2 + 4 3 = 10.2 3 = 3.4 Therefore, the upper sum of the function for three rectangles is 3.4 _ . 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 three rectangles as shown below. Refer to Figure (1). Take the minimum height of the first rectangle’s f ( x 0 ) value as 1.1, the minimum height of the second rectangle’s f ( x 1 ) value as 1, and the minimum height of the third rectangle’s f ( x 2 ) value as 1.1. Substitute 3 for n , 1.1 for f ( x 0 ) , 2 3 for Δ x , 1 for f ( x 1 ) , and 1

9th Edition

ISBN: 9780357631478

Author: James Stewart, Daniel K. Clegg, Saleem Watson

Publisher: Cengage Learning

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Chapter 5.1, Problem 8E

To determine

To evaluate:

The upper and lower sums for the function with n=3.

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

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Chapter 5.1, Problem 7E

Chapter 5.1, Problem 9E

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Chapter 5 Solutions

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The upper and lower sums for the function with n = 3 . The upper and lower sums for the function with n = 4 .

Solution Summary: The author evaluates the upper and lower sums for the function with n=3.

To evaluate:

The upper and lower sums for the function with n=3.

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

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Chapter 5 Solutions

The upper and lower sums for the function with n = 3 . The upper and lower sums for the function with n = 4 .

Solution Summary: The author evaluates the upper and lower sums for the function with n=3.

To evaluate: The upper and lower sums for the function with n=3. The upper and lower sums for the function with n=4.

Want to see the full answer?

Chapter 5 Solutions

To evaluate:

The upper and lower sums for the function with n=3.

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