Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.3* The Natural Exponential Function - 6.3* Exercises: 95

Answer

4.644

Work Step by Step

First step is to find the point of intersection of curves $y=e^{x},y=e^{3x}$ and $x=1$ . For this, $e^{x}=e^{3x}$ $x=3x$ $2x=0$ $x=0$ Thus, the point of intersection for given curves is (0, 1). Let A be the area of the region bounded by the curves, which is calculated as follows: $A=\int_ {0}^{1}(e^{3x}-e^{x})dx$ $=\int_ {0}^{1}e^{3x}dx-\int_ {0}^{1}e^{x}dx$ $=[\frac{1}{3}e^{3x}-e^{x}]_{0}^{1}$ $\approx 4.644$ Hence, the area of the region bounded by the curves is A =4.644 .
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