The closed surface consists of two parts: , the upper half of the ellipsoid; and , the interior and boundary of the ellipse in the plane .
The calculation was implemented in Example 9.9.6, where was called .
All that is needed here is to show that , so begin by calculating
=
A downward (and hence outward) unit normal on is clearly . Consequently,
To integrate this over the interior of , express in polar coordinates by the calculation
from which it follows that . Change to polar coordinates to obtain for the integral