(a) Explanation Given information: Take the curve as y = f ( x ) . Consider that the region lies between x = 0 and x = 10 . Hence, the limits are a = 0 and b = 10 . Consider the number of rectangles as n = 5 . Use the right end points of the curve as the lower estimate and the left end points of the curve as the upper estimate for decreasing behavior of the curve ( f ). The expression to find the lower estimate of the areas of n rectangles ( R n ) is shown below: R n = f ( x 1 ) Δ x + f ( x 2 ) Δ x + ... + f ( x n ) Δ x (1) Here, the height of first rectangle is f ( x 1 ) , the width is Δ x , the height of the second rectangle is f ( x 2 ) , and the height of n th rectangle is f ( x n ) . Draw five rectangles using right endpoints as shown in Figure (1). Find the width ( Δ x ) using the relation: Δ x = b − a n (2) Here, the upper limit is b , the lower limit is a , and the number of rectangles is n . Substitute 10 for b , 0 for a and 5 for n in Equation (2). Δ x = 10 − 0 5 = 2 Refer to Figure (1), Take the height of the first rectangle’s f ( x 1 ) value as 3.2, the height of the second rectangle’s f ( x 2 ) value as 1.8, the height of the third rectangle’s f ( x 3 ) value as 0.8, the height of the fourth rectangle’s f ( x 4 ) value as 0.2, and the height of the fifth rectangle’s f ( x 5 ) value as 0. Substitute 5 for n , 3.2 for f ( x 1 ) , 2 for Δ x , 1.8 for f ( x 2 ) , 0.8 for f ( x 3 ) , 0.2 for f ( x 4 ) , and 0 for f ( x 5 ) in Equation (1). R 5 = ( 3.2 × 2 ) + ( 1.8 × 2 ) + ( 0 (b) To determine The lower estimate and upper estimate of the area using ten rectangles.

Student Solutions Manual, Chapters 1-11 for Stewart's Single Variable Calculus, 8th (James Stewart Calculus)

8th Edition

ISBN: 9781305271814

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 4.1, Problem 1E

(a)

To determine

The lower estimate and upper estimate ofthe area under a graph using five rectangles.

(b)

To determine

The lower estimate and upper estimate of the area using ten rectangles.

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Chapter 3, Problem 24P

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Chapter 4 Solutions

Student Solutions Manual, Chapters 1-11 for Stewart's Single Variable Calculus, 8th (James Stewart Calculus)

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