Answer
\[\text{Does not exist}\]
Work Step by Step
\[\begin{align}
& f\left( x \right)={{\left( x-1 \right)}^{-1/4}},\text{ }\left( 1,2 \right] \\
& \text{From the graph shown below, we can calculate the volume} \\
& \text{using the Disk Method about the }x\text{-Axis }V=\int_{a}^{b}{\pi f{{\left( x \right)}^{2}}dx,\text{ }} \\
& V=\int_{0}^{\pi /2}{\pi {{\left[ \tan x \right]}^{2}}dx} \\
& V=\pi \int_{0}^{\pi /2}{{{\tan }^{2}}xdx} \\
& \text{The integrand is not defined for }x=\pi /2,\text{ then} \\
& V=\pi \underset{b\to \pi /{{2}^{-}}}{\mathop{\lim }}\,\int_{0}^{b}{{{\tan }^{2}}xdx} \\
& V=\pi \underset{b\to \pi /{{2}^{-}}}{\mathop{\lim }}\,\int_{0}^{b}{\left( {{\sec }^{2}}x-1 \right)dx} \\
& \text{Integrating} \\
& V=\pi \underset{b\to \pi /{{2}^{-}}}{\mathop{\lim }}\,\left[ \tan x-x \right]_{0}^{b} \\
& V=\pi \underset{b\to \pi /{{2}^{-}}}{\mathop{\lim }}\,\left[ \left( \tan b-b \right)-\left( \tan 0-0 \right) \right] \\
& V=\pi \underset{b\to \pi /{{2}^{-}}}{\mathop{\lim }}\,\left[ \left( \tan b-b \right) \right] \\
& \text{Evaluate the limit} \\
& V=\pi \left[ \left( \tan \frac{\pi }{2}-\frac{\pi }{2} \right) \right] \\
& V=\pi \left( \infty -\frac{\pi }{2} \right) \\
& V=\infty \\
& \text{The integral diverges, the volume does not exist} \\
\end{align}\]
