I would change to polar coordinates in the xz-plane. That is, , , with
Now the "position vector" for any point on that paraboid would be . The derivatives are and .
The cross product of those vectors is
so the "vector differential of surface area" is
But since we want "oriented in the positive y direction" we need to multiply throught by -1 to make the y-component positive:
(That is the same as swapping the two derivative vectors in the cross product.)
Since , and there is circular symmetry on the xz-plane, the limits of integration are r= 0 to 3 and to .