Integrating over a tetrahedron Let $S$ be the tetrahedron in $\mathbb{R}^3$ having vertices $(0,0,0), (0,1,2), (1,2,3), (-1,1,1)$. Calculate $\int_S f$ where $f(x,y,z) = x + 2y - z$. 
Before I show you guys what I have tried, please no solutions. Just small hints. Now, I have been trying to set up the integral by looking at $x$ being bounded between certain planes, etc. I ended up with $$\int_0^{x+2} \int_{\frac{z}{2} - \frac{x}{2}}^2 \int_{2y - z}^{3z - 4y} f\:\: \mathrm{d}x\mathrm{d}y\mathrm{d}z.$$ But this doesn't seem correct. The question came with a hint: To find a linear diffeomorphism $g$ to use as a change of variables, but I have been unable to find such a mapping between $S$ and the unit cube.

Hint to your hint: You can find a linear transformation sending the vectors $(0,0,0)(0,1,2)$ etc to the standard basis (actually, it is easiest to compute the inverse to this first). A linear transformation will change area in a uniform way (think about where arbitrary little cubes are sent), and the bounds after applying this transformation will be easier. (The keyword for all of this is determinants.)
