1) Integrate in the order dz dx dy.

z-integral terminals: Lower terminal is z = - sqrt{9 - x^2 - y^2} (lower surface). Upper terminal is z = sqrt{9 - x^2 - y^2} (upper surface).

The double integral dx dy is over the region of the xy-plane defined by x^2 + y^2 = 4. I'm sure you can set up the integral terminals. Personally, I'd switch to polar coordinates once the z-integration has been done .....

Note: You can use the symmetry of the problem .... Two times the triple integral where the lower terminal is z = 0 (lower surface) and the upper terminal is z = sqrt{9 - x^2 - y^2} (upper surface) ......