Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.5 Partial Fractions - 7.5 Exercises: 60

Answer

\[V = \frac{\pi }{2}\ln \,\left( 3 \right)\]

Work Step by Step

\[\begin{gathered} \hfill \\ We\,\,have\,\,to\,find\,the\,\,volume\,\,of\,\,the\,\,solid\,\,obtained\,\,by \hfill \\ revolving\,\,the\,\,region\,\,by\,\,\,y = \frac{1}{{4 - {x^2}}}\,\,,\,\,y = 0\,\,,\,\,x = - 1\,\,\,and\,\, \hfill \\ x = 1\,\,is\,\,revolved\,\,about\,\,x - \,axis. \hfill \\ \hfill \\ Applying\,\,the\,\,\,disk\,\,method \hfill \\ \hfill \\ V = \pi \int_{ - 1}^1 {\frac{1}{{1 - {x^2}}}} dx\,\, \hfill \\ \hfill \\ by\,\,symmetry \hfill \\ \hfill \\ V = 2\pi \int_0^1 {\frac{1}{{4 - {x^2}}}dx} \hfill \\ \hfill \\ integrate\,\,and\,\,evaluate \hfill \\ \hfill \\ V = 2\pi \,\,\left[ {\frac{1}{4}\ln \left| {\frac{{2 + x}}{{2 - x}}} \right|} \right]_0^1 \hfill \\ \hfill \\ solution \hfill \\ \hfill \\ V = \frac{\pi }{2}\ln \,\left( 3 \right) \hfill \\ \end{gathered} \]
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