Calculate
where C is the curve obtained by intersecting surfaces and , oriented so that the z-coordinate increases along C.
That C has to be "oriented so that the z-coordinate increases along C" is a ridiculous requirement that cannot be satisfied generally, because C is a closed curve. So moving along C we may see the z-coordinates increase, only to see them necessarily decrease later on.
is a circle with center at the origin and radius 2. A standard parmeterization is . , then, becomes .
That is, a parameterization of the path is , , .
However, as veritaserum2002 said, in going around that curve, in either direction, the z value, as well as both x and y values, both increase and decrease so "oriented so that the z-coordinate increases along C" makes no sense.
The "positive orientation" for this curve, the orientation so that the "oriented surfaces" (normal axis pointing in the positive z direction) were "on the left" as you move around the curve, is with going from 0 to