Hey xl5899.
I think you may have the wrong region.
If you look at your diagram you have two regions [look at the xy plane for the moment and forget the z-plane].
You have two areas - one on each side of the line y = x^2.
Try drawing a diagram and do two sets of limits for each area [i.e. each side of that line] and verify that you are indeed integrating over the correct region of area.
If you aren't then the answer will be significantly different and I have a "hunch" you are integrating the wrong area.
The z-values won't matter because the function is [in this case] independent of the z-value and it should just be multiplied by the height [just like a cylinder is multiplied by the height after you get the area of the cross sectional circle].