Yes, you have to parameterize the paraboloid surface. And looking at that , I would think of using "cylindrical coordinates" but with the polar coordinates part in the yz-plane rather than the xy-plane. That is, take x as a parameter and as the other parameter with , so that as required.
The condition that the surface be inside the cylinder just means that we must have . Your integral will be for from 0 to , x from 0 to 9.