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 .