Answer
$$V = \frac{{128\pi }}{7}$$
Work Step by Step
$$\eqalign{
& {\text{From the graph of the region shown below}} \cr
& f\left( x \right) = {x^3}{\text{ on the interval }}\left[ {0,2} \right] \cr
& {\text{Using the disk method about the }}x{\text{ - axis}} \cr
& V = \int_a^b {\pi f{{\left( x \right)}^2}} dx \cr
& V = \int_0^2 {\pi {{\left( {{x^3}} \right)}^2}} dx \cr
& V = \pi \int_0^2 {{x^6}} dx \cr
& {\text{Integrating}} \cr
& V = \pi \left[ {\frac{1}{7}{x^7}} \right]_0^2 \cr
& V = \frac{\pi }{7}\left[ {{{\left( 2 \right)}^7} - {{\left( 0 \right)}^7}} \right] \cr
& {\text{Simplifying}} \cr
& V = \frac{{128\pi }}{7} \cr} $$
