How do you find the volume of the solid generated by revolving the region bounded by the curves #y=2x^2#; y=0; x=2 rotated about the x-axis?

Question

2149 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-10-14T01:23:02

See the explanation below.

Here is the region and a thin slice taken perpendicular to the axis of rotation. (So we'll be using disks.)

enter image source here

The representative disk (at #x#) has volume #pir^2*"thickness"#

In this case, radius #r = 2x^2# and thickness = #dx#.

#x# varies from #0# to #2#, so we need

#int_0^2 pi(2x^2)^2 dx = 4pi int_0^2 x^4 dx# which is straightforward to evaluate.