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