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?

1 Answer
bp
Jul 29, 2015

#pi/30#

Explanation:

The curve represents a horizontal parabola as seen in the picture

enter image source here

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#

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