Integrating z over a tetrahedron bounded by x+y+z=1

[Alem2000]

the question is "evaluate $$\iiint z \,dv$$, of a solid tetrahedron bounded by the four planes x=0,y=0,z=0, and x+y+z=1"
I can set up the problem correctly but i can't seem to integrate it right
$$\int_{0}^1 \int_{0}^{1-x} \int_{0}^{1-x-y} z dzdydx$$
$$(1/2) \int_{0}^1 \int_{0}^{1-x} (1-x-y)^2dydx$$
can someone please show me the last few steps of this problem?

 

[dextercioby]

Open the brackets and integrate each term with corresponding limits...

Daniel.

 

[arildno]

For the evaluation of the inner integral, use the y-anti-derivative :
$$-\frac{1}{3}(1-x-y)^{3}$$