Question

How do you find the volume of the solid obtained by rotating the region bounded by the curves `y=2x^2+5`, and `y=x+3` and the y-axis, and `x=3` rotated around the x axis?

Answer 1

Use disks/washers.

Explanation:

Sketch the region. Note that `2x^2+5` is above (greater than) `x+3`, so the parabola is farther from the axis of rotation.

Therefore:
At a particular `x`, the large radius is, `R = 2x^2+5`, and the small radius is `r = x+3`. The thickness of the disks is `dx`

The volume of each representative disk would be `pi * "radius"^2 * "thickness"`. So the large disk has volume: `pi(2x^2+5)^2 dx` and the small one has volume `pi (x+3)^2 dx`

The volume of the washer is the difference, or `piR^2dx-pir^2dx`and the resulting solid has volume:

`V = pi int_0^3 ((2x^2+5)^2 - (x+3)^2) dx`

`= pi int_0^3 (4x^4+19x^2-6x+16) dx`

You can finish the integral to get `(1932 pi)/5`