Integral of $f(x,y,z)=x+2y+z$ over a tetrahedron.

Question

Let $S$ be the tetrahedron in $\mathbb{R}^{3}$ having vertices $(0,0,0),(1,2,3), (0,1,2)$ and $(-1,1,1)$. Evaluate $\int_{S} f$, where $f(x,y,z)=x+2y-z$.

Answer 1

Hint: Triangles, tetrahedra, and its higher dimensional brethren have a notion of being able to combine points inside them (in fancy words, they are closed under convex combinations). The upshot of this is that if $\mathbf x,\mathbf y,$ and $\mathbf z$ are vertices of a triangle, then every point inside them may be written as $a\mathbf x+b\mathbf y+c\mathbf y$ for some $0\leq a,b,c\leq 1$ such that $a+b+c=1$. Note that this means there are only two independent variables; once $a$ and $b$ are chosen, then we are forced to have$c=1-(a+b)$.

Extend this idea to a tetrahedron, and use it to rewrite your integral. The limits you end up with should be very simple.