See the image for a representative cylindrical shell. The total volume of the solid of revolution is the sum of an infinite number of such infinitesimally thin shells. Each shell has a surface area determined exactly by the value of [tex]x[/tex], with radius [tex]3-x[/tex] and height [tex]8-x^3[/tex]. We have [tex]x[/tex] ranging from 0 to 2, since [tex]y=x^3=8\implies x=2[/tex].
So the volume is given by the integral
[tex]\displaystyle2\pi\int_0^2(3-x)(8-x^3)\,\mathrm dx=\frac{264\pi}5[/tex]
