Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 8 - Further Techniques and Applications of Integration - 8.4 Improper Integrals - 8.4 Exercises - Page 452: 32

Answer

Divergent

Work Step by Step

\[\begin{align} & f\left( x \right)={{e}^{-x}},\text{ }\left( -\infty ,e \right] \\ & \text{The area under the curve is given by } \\ & A=\int_{-\infty }^{e}{{{e}^{-x}}}dx \\ & \text{By the definition of improper integrals} \\ & A=\underset{a\to -\infty }{\mathop{\lim }}\,\int_{a}^{e}{{{e}^{-x}}}dx \\ & A=\underset{a\to -\infty }{\mathop{\lim }}\,\left[ {{e}^{-x}} \right]_{a}^{e} \\ & A=\underset{a\to -\infty }{\mathop{\lim }}\,\left[ {{e}^{-e}}-{{e}^{-a}} \right] \\ & \text{Evaluate when }a\to -\infty . \\ & A={{e}^{-e}}-{{e}^{-\left( -\infty \right)}} \\ & A={{e}^{-e}}-{{e}^{\infty }} \\ & A=\infty \\ & \text{The improper integral diverges.} \\ & \text{Divergent} \\ \end{align}\]
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