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: 14

Answer

$$1$$

Work Step by Step

$$\eqalign{ & \int_0^\infty {50{e^{ - 50x}}} dx \cr & {\text{by the definition of an improper integral}}{\text{}} \cr & \int_0^\infty {50{e^{ - 50x}}} dx = \mathop {\lim }\limits_{b \to \infty } \int_0^b {50{e^{ - 50x}}} dx \cr & {\text{write as}} \cr & = - \mathop {\lim }\limits_{b \to \infty } \int_1^b {50{e^{ - 50x}}} \left( { - 50} \right)dx \cr & {\text{integrating by using }}\int {{e^u}} du = {e^u} + C \cr & = - \mathop {\lim }\limits_{b \to \infty } \left[ {{e^{ - 50x}}} \right]_0^b \cr & {\text{ using ftc}} \cr & = - \mathop {\lim }\limits_{b \to \infty } \left[ {{e^{ - 50b}} - {e^{ - 50\left( 0 \right)}}} \right] \cr & = - \mathop {\lim }\limits_{b \to \infty } \left( {{e^{ - 50b}} - 1} \right) \cr & {\text{evaluating the limit when }}b \to \infty \cr & = - \left( {{e^{ - \infty }} - 1} \right) \cr & = - \left( {0 - 1} \right) \cr & = 1 \cr} $$
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