Decompose the surface into three components,
, corresponding respectively to the cylindrical region and the top and bottom disks:
where
and
,
where
and
, and
where
and
.
For the cylinder, we have
and the integral over this surface is
Bottom disk:
and the integral over the bottom disk is
The setup for the integral along the top disk is similar to that for the bottom disk, except that
:
Finally, the value of the integral over the entire surface is the sum of the integrals over the component surfaces: