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CalculusCalculus (MindTap Course List)8th Edition

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Chapter 4.1, Problem 27E
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Let A be the area under the graph of an increasing continuous function f from a to h, and let Ln and Rn be the approximations to A with n subintervals using left and right endpoints, respectively.

(a) How are A, Ln, and Rn related?

(b) Show that

R n − L n = b − a n [ f ( b ) − f ( a ) ]

Then draw a diagram to illustrate this equation by showing that the n rectangles representing Rn - Ln can be reassembled to form a single rectangle whose area is the right side of the equation.

(c) Deduce that

R n − A < b − a n [ f ( b ) − f ( a ) ]

To determine

a)

To describe:

A relation

Explanation

1) Concept:

The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles.

A=limn→∞⁡Rn=limn→∞fx1∆x+fx2∆x+…+fxn∆x=limn→∞∑i=1nfxi∆x

The width of the interval a, b is (b-a)

To determine

b)

To show:

Rn-Ln=b-anfb-fa

To determine

c)

To deduce:

Rn-A<b-anfb-fa

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