Surface integral: Parameterize the closed surface by
with
and
, where
defines the paraboloid part and
the planar part of the total surface
.
We have
so we get
The second integral vanishes when computing the dot product, so we're left with the first integral which reduces to
Volume integral (divergence theorem): We have divergence
By the divergence theorem, the flux is equivalent to the volume integral
where
denotes the space enclosed by the surface
. Converting to cylindrical coordinates lets us write the integral as
as desired.