Question

How do you use the disk method to find the volume of the solid formed by rotating the region bounded by `y = 2x` and `y = x^2` about the y-axis?

Answer 1

`y = 2x` and `y = x^2` intersect at `(0,0)` and at `(2,4)`

I can't get both curves on one graph, but I'll assume you can graph the line and the parabola.

Rotating around the `y` axis and using disks, means our independent variable will be `y`.

The representative slice is horizontal with left end (little radius: r) on the line `x=1/2y` and right end (big radius: R) on the parabola `x=sqrty`.
The thickness of the disk will be `dy`, and the limits of integration will be `0` to `4`.

`int_0^4 piR^2-pir^2 dy = piint_0^4 ((sqrty)^2-(y/2)^2) dy`

`pi int_0^4 (y-y^2/4) dy = pi[y^2/2-y^3/12]_0^4 =pi[8-16/3] =(8pi)/3`