I was wondering, is it possible to find the volume under a parameterized surface above a domain? The domain is a parameterized area on the xy-plane.
My thoughts: Maybe you could multiply the determinant of the matrix of the partial derivatives of the domain. Then you could multiply by the height of the graph over that area (the third component of the parameterized surface). Finally I could double integrate with respect to the two variables of the area.
The problem is: I do not know how to make this idea concrete, or if it is even correct.
If you actually know that the surface is a graph of some function $f$ over some domain $D$ in the target space of your parametriziation you need to find that function. The volume under the surface is then just the integral of $f$.