Split up the surface into three main components , where
is the region in the plane bounded by ;
is the piece of the cylinder bounded between the two planes and ;
and is the part of the plane bounded by the cylinder .
These surfaces can be parameterized respectively by
where and ,
where and ,
where and .
The surface integral of a function along a surface parameterized by is given to be
Assuming we're just finding the area of the total surface , we take , and split up the total surface integral into integrals along each component surface. We have