Explanation Given: Show that if f(x) is positive and monotonic, then the area A under its graph over [a, b] satisfies | R N − A | ≤ ( b − a ) N | f ( b ) − f ( a ) | . Calculation: Let f(x) be continuous, positive, and monotonic on [a, b]. Let A be the area between the graph of f and the x -axis over [a, b]. For specificity, say f is increasing, we have L N ≤ L ≤ R N . By exercise 88 and the fact that A lies between L N and R N , we therefore have 0 ≤ R N − A ≤ R N − L N = ( b − a ) N ( f ( b ) − f ( a ) ) ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 5.1, Problem 88E

To determine

The area A under its graph over [a, b] satisfies $| R_{N} - A | \leq \frac{(b - a)}{N} | f (b) - f (a) | .$

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Chapter 5.1, Problem 87E

Chapter 5.1, Problem 89E

Chapter 5 Solutions

Calculus - Standalone book

Ch. 5.1 - Prob. 1PQ Ch. 5.1 - Prob. 2PQ Ch. 5.1 - Prob. 3PQ Ch. 5.1 - Prob. 4PQ Ch. 5.1 - Prob. 5PQ Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E

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The area A under its graph over [a, b] satisfies | R N − A | ≤ ( b − a ) N | f ( b ) − f ( a ) | .

Solution Summary: The author explains that if f(x) is continuous, positive, and monotonic, then the area A under its graph over [a, b] satisfies.

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The area A under its graph over [a, b] satisfies $| R_{N} - A | \leq \frac{(b - a)}{N} | f (b) - f (a) | .$

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Chapter 5 Solutions

The area A under its graph over [a, b] satisfies | R N − A | ≤ ( b − a ) N | f ( b ) − f ( a ) | .

Solution Summary: The author explains that if f(x) is continuous, positive, and monotonic, then the area A under its graph over [a, b] satisfies.

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The area A under its graph over [a, b] satisfies |RN−A|≤(b−a)N|f(b)−f(a)|.

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Chapter 5 Solutions

The area A under its graph over [a, b] satisfies $| R_{N} - A | \leq \frac{(b - a)}{N} | f (b) - f (a) | .$