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

1 Answer
Oct 26, 2015

See the explanation section below.

Explanation:

Here is a picture of the region with a representative slice taken perpendicular to the axis of rotation. This is a set-up to use disks to find the volume. The thickness is #dy#

enter image source here

As #y# varies from #0# to #1#, the disk at #y# has radius #y-y^2# and thickness #dy#.

The volume of the representative disk is
#pi xx "radius"^2xx d"thickness" = pi(y-y^2)^2 dy##.

So, to find the volume of the solid, we need to evaluate

#int_0^1 pi(y-y^2)^2 dy = piint_0^1 (y^2-2y^3+y^4) dy#.

This integral evaluates to #pi/30#.

Bonus

Here is a link to the same volume problem using shells instead of disks.

http://socratic.org/questions/how-do-you-use-the-shell-method-to-set-up-and-evaluate-the-integral-that-gives-t-38#179893