Consider a surface described by the $n^{th}$-order bivariate polynomial equation
$$P_n(x,y) = \sum_{i=0}^n\sum_{j=0}^{n-i} c_{ij}x^iy^j, $$
and a tetrahedron that intersects this surface. For simplicity, I'm only considering the following two cases - the surface only intersects the tetrahedron at $m$ edges and the $m$ faces connected to those edges, where $m\in\{3,4\}$ (I am interested in all possible intersection cases, but I do not want to over-complicate the question). Is there a means of applying calculus/differential geometry to analytically compute - (i) the surface area of the intersection between the surface and a tetrahedron and (ii) the volume of intersection between the region below the polynomial surface and the tetrahedron - for the aforementioned two cases?