How do you find the volume of the solid if the region in the first quadrant bounded by the curves #x=y-y^2# and the y axis is revolved about the y axis?

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Answer 1 · 2015-05-07T20:08:39

I've edited your question to what I think you meant to ask. If I'm mistaken, please accept my apologies.

The curve #x=y-y^2# is a parabola that opens to the left. It has #y# intercepts #0# and #1#.

Using discs, we see that the radius is #x# or #y-y^2#, the thickness is #dy#, so the volume or a representative disc is

#pir^2 d9y = pi (y-y^2)^2 dy#

Integrate from #y=0# to #y=1#

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

Which I believe is #(17 pi)/60#