For each [tex]x[/tex] in the interval [tex]0\le x\le2[/tex], the corresponding shell has radius [tex]x+9[/tex] (horizontal distance from [tex]x[/tex] to the rotation axis) and height [tex]8-x^3[/tex] (vertical distance between [tex]y=8[/tex] and [tex]y=x^3[/tex]). A shell of radius [tex]r[/tex] and height [tex]h[/tex] has area [tex]2\pi rh[/tex], so the volume is
[tex]\displaystyle2\pi\int_0^2(x+9)(8-x^3)\,\mathrm dx=2\pi\int_0^272+8x-9x^3-x^4\,\mathrm dx=\boxed{\frac{1176\pi}5}[/tex]
