Consider the closed region 
bounded simultaneously by the paraboloid and plane, jointly denoted 
. By the divergence theorem,
And since we have
the volume integral will be much easier to compute. Converting to cylindrical coordinates, we have
Then the integral over the paraboloid would be the difference of the integral over the total surface and the integral over the disk. Denoting the disk by
, we have
Parameterize by
which would give a unit normal vector of
. However, the divergence theorem requires that the closed surface be oriented with outward-pointing normal vectors, which means we should instead use 
.
Now,
So, the flux over the paraboloid alone is
And since we have
the volume integral will be much easier to compute. Converting to cylindrical coordinates, we have
Then the integral over the paraboloid would be the difference of the integral over the total surface and the integral over the disk. Denoting the disk by
Parameterize
which would give a unit normal vector of
Now,
So, the flux over the paraboloid alone is