it looks good-however the orientatation seems a little strange.
you have a parabaloid aligned along the positive y axis
Usually we specify inward or outward--here on the half in the first and 8th octants you have an outward normal but in the 2d and 5th you have an inward normal.
So is this correct?
and of course your region of integration is a circle of radius 6 in the x-z plane