Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis. y = x3, y = 8, x = 0; about x = 9 V =

Answer 1

For each $x$ in the interval $0\le x\le2$, the corresponding shell has radius $x+9$ (horizontal distance from $x$ to the rotation axis) and height $8-x^3$ (vertical distance between $y=8$ and $y=x^3$). A shell of radius $r$ and height $h$ has area $2\pi rh$, so the volume is

$\displaystyle2\pi\int_0^2(x+9)(8-x^3)\,\mathrm dx=2\pi\int_0^272+8x-9x^3-x^4\,\mathrm dx=\boxed{\frac{1176\pi}5}$