Explanation Given information: The equation of the graph is y = 4 e − 4 x for x ≥ 0 . Explanation: Definition: Let a be fixed and suppose that f ( x ) is a non-negative function for x ≥ a , and if lim b → ∞ ∫ a b f ( x ) d x = L , then ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x = L First, write the improper integral using the given information and definition. By using the definition, The area defined by an improper integral is ∫ 0 ∞ 4 e − 4 x d x Let b > 0 , then lim b → ∞ ∫ 0 b 4 e − 4 x d x . Now solve improper integral as follows: By using the constant multiple rule and the exponential rule, ∫ 0 b 4 e − 4 x d x = 4 ( − e − 4 x 4 | 0 b ) = 4 4 ( − e − 4 x | 0 b ) = ( − e − 4 b − ( − e − 4 ( 0 ) ) ) </

15th Edition

ISBN: 9780137590438

Author: Goldstein

Publisher: PEARSON

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Question

Chapter 9.6, Problem 16E

To determine

The area under the graph of y=4e− 4x for x≥ 0

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Chapter 9.6, Problem 15E

Chapter 9.6, Problem 17E

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Chapter 9 Solutions

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The area under the graph of y = 4 e − 4 x for x ≥ 0

Solution Summary: The author explains that the area under the graph of y=4e-4x is 1. Write the improper integral using the given information and definition.

The area under the graph of y=4e− 4x for x≥ 0

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Chapter 9 Solutions