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Check the picture below.
we know the lines are angle bisectors, so the line makes twin angles, so the line FP is making twin angles, we also know the angle FZP and the angle FYP are right-angles, as well as FP is a common hypotenuse to two right triangles, thus by the HA postulate for right triangles, triangle FPZ and triangle FPY are both congruent, so their sides are also congruents, thus PY = PZ.
I've attached a plot of one such cross-section (orange) over the region in the x-y plane (blue), including the bounding curves (red). (I've set 
for this example.)
The length of each cross section (the side lying in the base) has length determined by the horizontal distance
between the y-axis 
and the curve 
. In terms of 
, this distance is 
. The height of each cross section is twice the value of , so the area of each rectangular cross section should be 
.
This means the volume would be given by the integral
The length of each cross section (the side lying in the base) has length determined by the horizontal distance
This means the volume would be given by the integral