Question

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

Answer 1

`pi/30`

Explanation:

The curve represents a horizontal parabola as seen in the picture

The region rotated about y axis is the shaded region.

The volume of the solid so generated would be(consider an elementary strip of length and thickness `delta`y. If it is rotated about x axis its volume would be `pix^2 dy`. The volume of the solid generated by rotating the whole shaded region would be

`int_0^1 pi x^2 dy`

=`int_0^1 pi (y-y^2)^2 dy`

=`int_0^1 pi(y^2 -2y^3 +y^4)dy`

=`pi(y^3 /3 -2y^4 /4 +y^5 /5)_0^1`

=`(pi)/30`