

By the divergence theorem, the flux of

across the *closed* surface

combined with the plane

is given by a volume integral over the closed region:

So in fact, to find the flux over

alone, we'll need to subtract the flux of

over the planar portion, oriented outward. First, compute the volume integral by converting to cylindrical coordinates:



If the surface does actually contain

, then you can stop here; otherwise, continue.
Now, parameterize the part of the *closed* surface in

by

where

and

. We get a surface element

We don't need to worry about the first two components of
and so the surface integral over this region is

Then the total flux over

alone is

.