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