Question

How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by `y=x^3`, x=1, y=0 revolved about the y-axis?

Answer 1

Please see below.

Explanation:

A representative shell has volume

`V = 2pirh * "thickness" = 2pi(x)(y)dx = 2pix^4 dx`

`x` varies from `0` to `1`, so the solid has volume:

`V = int_0^1 2pix^4 dx = (2pi)/5`