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?

1 Answer
Jun 21, 2015

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^2dxand 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