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}`