The problem didn't ask for the area. It asked for the volume.

I am confused about the conditions set in the original problem. You said:

y=x-x^2
y=0

I think you need a range of y's. Otherwise, the volume would be zero because the solid would only have 2 dimensions (x and z). It's got to have a range of y's to give it 3 dimensional volume. Does that make sense? I may have missed the point.