Calculus: Early Transcendentals 9th Edition

Published by Cengage Learning
ISBN 10: 1337613924
ISBN 13: 978-1-33761-392-7

Chapter 6 - Section 6.3 - Volumes by Cylindrical Shells - 6.3 Exercises - Page 465: 11

Answer

$V = 6\pi $

Work Step by Step

$$\eqalign{ & y = \frac{1}{x}{\text{, }}y = 0,{\text{ }}x = 1,{\text{ }}x = 4 \cr & {\text{Apply the shell method about the }}y{\text{ - axis}}{\text{.}} \cr & V = \int_a^b {2\pi x} f\left( x \right)dx,{\text{ where }}0 \leqslant a \leqslant b \cr & {\text{Therefore}}{\text{,}} \cr & V = \int_1^4 {2\pi x\left( {\frac{1}{x}} \right)} dx \cr & V = 2\pi \int_1^4 {dx} \cr & {\text{Integrating}} \cr & V = 2\pi \left[ x \right]_1^4 \cr & {\text{Evaluating}} \cr & V = 2\pi \left[ {4 - 1} \right] \cr & V = 6\pi \cr & \cr} $$
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