If you rewrite your equation like this : x²+y²=A-z, you see that each "horizontal" section of the paraboloid is a circle where r²=A-z. The area of a quarter of this circle ("first octant") is 𝜋(A-z)/4. Integrating this between 0 and A, you get 𝜋A²/8 which must be equal to 32𝜋. So A=16. Right ?

If you rewrite your equation like this : x²+y²=A-z, you see that each "horizontal" section of the paraboloid is a circle where r²=A-z. The area of a quarter of this circle ("first octant") is 𝜋(A-z)/4. Integrating this between 0 and A, you get 𝜋A²/8 which must be equal to 32𝜋. So A=16. Right ?