When I solving the practice exercise problems at the end of the section, I stumbled upon this problem, which I have been trying to figure out how to compute the integral, but couldn't. Can someone please help?
Integrate $G(x,y,z) = xyz$ over the triangular surface with vertices $(1,0,0),\,(0,2,0)$ and $(0,1,1)$.
What I have tried:
I found the surface in question: $f(x,y,z) = 2x+y+z = 2$
If you have $3$ vertices $A$, $B$ and $C$, find a parametrization for the triangle, like: $$ u = B-A\\ v = C-A\\ (x,y,z) = A+tu+kv,\quad\text{where }t\in [0,1-k], k\in[0,1] $$ The differential area $d\Omega$ is $||u\times v||\,dt\,dk$. Then: $$ \iint_\triangle G(x,y,z)\,d\Omega = \int_0^1\int_0^{1-k} G(A+tu+kv)\,||u\times v||\,dt\,dk $$