Question

Find the volume of the solid that results when the region enclosed by the curves y=1+x^3 x=1 and y=9 is revolved about the y-axis

Answer 1

The volume is given by the integral


`\displaystyle2\pi\int_1^2x(9-(1+x^3))\,\mathrm dx=2\pi\int_1^2(8x-x^4)\,\mathrm dx`

with the shell method, and


`\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`

with the washer method. Both integrals give a volume of
`\frac{58\pi}5`.