Couldn't stop thinking about this calculation. The line integral on the right in the Divergence form of Green's theorem is actually the line integral of the normal component of <f,g> along a curve. Hence, this is a flux calculation, for which Maple has a built-in tool. So, using Kitonum's structure, it is possible to implement the line integral on the bounding SCROC with the Flux command in the Student VectorCalculus package. Just to prove to myself that this works, I shortened the code to the following, and verified that it gives the same answer(s) as the earlier code. By no means does this detract from the original insight exhibited by Kitonum.
IntOverDomain := proc(f, L)
local n, i, j, m, yk, yb, xk, xb, Q, p, P, R, V;
use Student:-VectorCalculus in
for i from 1 to n do
if type(L[i], listlist(algebraic)) then
add(P[i], i = 1 .. n);