Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 6 - Section 6.5 - Average Value of a Function - 6.5 Exercises - Page 463: 5



Work Step by Step

Given: Average value of the function $f(x)=e^{sint}cost$ over the interval $[0,\pi/2]$ is given by $f_{avg}=\frac{1}{\pi/2-0}\int_{0}^{\pi/2}e^{sint}costdt$ Substitute $sint=u$ and $costdt=du$ Limits of integration changes $\int_{0}^{\pi/2}$ to $\int_{sin0}^{sin\pi/2}=\int_{0}^{1}$ $f_{avg}=\frac{2}{\pi}\int_{0}^{1}e^{u}du$ $f_{avg}=\frac{2}{\pi}[e^{u}]_{0}^{1}$ Hence, $f_{avg}=\frac{2(e-1)}{\pi}$
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