Answer
\[V=\frac{72\left( {{2}^{1/3}} \right)\pi }{5}\]
Work Step by Step
\[\begin{align}
& f\left( x \right)={{\left( 4-x \right)}^{-1/3}},\text{ }\left[ 0,4 \right) \\
& \text{we can calculate the volume using the Shell Method the }y\text{-Axis } \\
& V=\int_{a}^{b}{2\pi xf\left( x \right)dx,\text{ }} \\
& V=\int_{0}^{4}{2\pi x{{\left( 4-x \right)}^{-1/3}}dx} \\
& V=\int_{0}^{4}{\frac{2\pi x}{{{\left( 4-x \right)}^{1/3}}}dx} \\
& \text{The integrand is not defined for }x=4,\text{ then} \\
& V=\underset{b\to {{4}^{+}}}{\mathop{\lim }}\,\int_{0}^{b}{\frac{2\pi x}{{{\left( 4-x \right)}^{1/3}}}dx} \\
& V=2\pi \underset{b\to {{4}^{+}}}{\mathop{\lim }}\,\int_{0}^{b}{\frac{x}{{{\left( 4-x \right)}^{1/3}}}dx} \\
& \\
& \text{Integrating} \\
& \int{\frac{x}{{{\left( 4-x \right)}^{1/3}}}}dx,\text{ }u=4-x,\text{ }x=4-u,\text{ }dx=-du \\
& =\int{\frac{4-u}{{{u}^{1/3}}}}\left( -du \right)=\int{\frac{u-4}{{{u}^{1/3}}}}du=\int{\left( {{u}^{2/3}}-4{{u}^{-1/3}} \right)du} \\
& =\frac{3}{5}{{u}^{5/3}}-6{{u}^{2/3}}+C \\
& =\frac{3}{5}{{\left( 4-x \right)}^{5/3}}-6{{\left( 4-x \right)}^{2/3}}+C \\
& \\
& V=2\pi \underset{b\to {{4}^{-}}}{\mathop{\lim }}\,\left[ \frac{3}{5}{{\left( 4-x \right)}^{5/3}}-6{{\left( 4-x \right)}^{2/3}} \right]_{0}^{b} \\
& V=2\pi \underset{b\to {{4}^{-}}}{\mathop{\lim }}\,\left[ \frac{3}{5}{{\left( 4-b \right)}^{5/3}}-6{{\left( 4-b \right)}^{2/3}} \right]-2\pi \underset{b\to {{4}^{-}}}{\mathop{\lim }}\,\left[ \frac{3}{5}{{\left( 4 \right)}^{5/3}}-6{{\left( 4 \right)}^{2/3}} \right] \\
& \text{Evaluate the limit} \\
& V=2\pi \underset{b\to {{4}^{-}}}{\mathop{\lim }}\,\left[ \frac{3}{5}{{\left( 4-4 \right)}^{5/3}}-6{{\left( 4-4 \right)}^{2/3}} \right]-2\pi \left[ \frac{3}{5}{{\left( 4 \right)}^{5/3}}-6{{\left( 4 \right)}^{2/3}} \right] \\
& V=-2\pi \left[ \frac{3}{5}{{\left( 4 \right)}^{5/3}}-6{{\left( 4 \right)}^{2/3}} \right] \\
& V=12\pi {{\left( 4 \right)}^{2/3}}-\frac{6\pi }{5}{{\left( 4 \right)}^{5/3}} \\
& V=12\pi {{\left( 4 \right)}^{2/3}}-\frac{6\pi }{5}{{\left( 4 \right)}^{5/3}} \\
& V=\frac{72\left( {{2}^{1/3}} \right)\pi }{5} \\
\end{align}\]
