TITLE: Help evaluating triple integral over tetrahedron
QUESTION [11 upvotes]: I have a triple integral of $\iiint xyz\,dx\,dy\,dz$  over the volume of a tetrahedron with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. 
Normally I would just have limits 0 to 1 but that does not seem to work. How do I solve a problem like this?

REPLY [4 votes]: I agree entirely with the accepted answers; that is how the integral should be attempted analytically. However, if you wish to compute integrals of this form over the unit tetrahedron efficiently, or even just check your answers, it is worth noting that there is an explicit formula for the result as a function of the powers.
$$\int_0^1 \int_0^{1-z} \int_0^{1-y-z} x^a y^b z^c \, dx \, dy \, dz = \frac{1}{3!} \frac{a! \, b! \, c!}{(a + b + c)!} \frac{1}{3 + a + b + c \choose 3} $$
See This mathoverflow question for proof and an extension to other dimensions (triangles, 4-D tetrahedra, etc...).