The volume is given by the integral
[tex]\displaystyle2\pi\int_1^2x(9-(1+x^3))\,\mathrm dx=2\pi\int_1^2(8x-x^4)\,\mathrm dx[/tex]
with the shell method, and
[tex]\displaystyle\pi\int_2^9((\sqrt[3]{y-1})^2-1^2)\,\mathrm dy=\pi\int_2^9((y-1)^{2/3}-1)\,\mathrm dy[/tex]
with the washer method. Both integrals give a volume of [tex]\dfrac{58\pi}5[/tex].
