I'm not sure what your B and C is but I'm going to guess.

The surface whose boundary is is given parametrically as

where (see attached graph). The surface is blue, your boundary is in black and I think the B and C you mention are the red lines in the graph one horizontal and one vertical. Now the line integral is easy. Introducing your parametric equations directly into the line integral gives

Now for the Stoke's part. If so

Further

so that