Question

Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by:

y = x^2 , y = 0 , and x = 2 , about the y -axis.

Answer 1

The volume is

`\displaystyle2\pi\int_0^2x^3\,\mathrm dx=\frac\pi2 x^4\bigg|_0^2=8\pi`

Each shell has a height of
`x^2` (distance between
`y=x^2` and
`y=0`) and width equal to the distance to the axis of revolution (
`x-0=x`), hence contributing a surface area of
`2\pi x^3`. Sum (integrate) over all
`x` in the region of interest to get the volume.