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: 59

Answer

\[\frac{{2\pi }}{3}\,\,\ln \,\,2\]

Work Step by Step

\[\begin{gathered} Apply\,\,\,the\,\,\,disk\,\,method\,\,for\,\,volume \hfill \\ \hfill \\ V = \pi \,\,\left[ {\int_1^2 {\,\left( {\frac{1}{{\,{{\left( {\sqrt {x\,\left( {3 - x} \right)} } \right)}^2}}}} \right)} dx} \right] \hfill \\ \hfill \\ simplify\,\,\frac{1}{{\,{{\left( {\sqrt {x\,\left( {3 - x} \right)} } \right)}^2}}} \hfill \\ \hfill \\ V = \pi \int_1^2 {\,\left( {\frac{1}{{\,\left( {x\,\left( {3 - x} \right)} \right)}}} \right)} dx \hfill \\ \hfill \\ decompose\,\,\frac{1}{{\,x\,\left( {3 - x} \right)}}{\text{ into partial fractions}} \hfill \\ \hfill \\ V = \frac{\pi }{3}\int_1^2 {\,\left( {\frac{1}{x} + \frac{1}{{3 - x}}} \right)} dx \hfill \\ \hfill \\ in\,tegrate\,\,and\,\,evaluate \hfill \\ \hfill \\ V = \frac{\pi }{3}\,\,\left[ {\ln \left| x \right| - \ln \left| {x - 3} \right|} \right]_1^2 \hfill \\ \hfill \\ V = \frac{\pi }{3}\,\,\left[ {\ln \left| 2 \right| + \ln \left| 2 \right|} \right] \hfill \\ \hfill \\ solution \hfill \\ \hfill \\ \frac{{2\pi }}{3}\,\,\ln \,\,2 \hfill \\ \end{gathered} \]
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