Multiple Integration non rectangular domain I'm trying to do the following question from an advanced calculus past paper at university:
Suppose that $f(x)$ is continuous in $0\le x  \le 1$ show that
$$\int_0^1 \int_x^1 \int_x^y f(x)f(y)f(z) dzdydx = \frac{1}{6} \left( \int_0^1 f(x) dx \right)^3$$ 
I've found an example in two dimensions (so answer is area square /2) but can't see how to apply the same method to 3 dimensions. Not sure it's correct, but I get down to.
$$I=\frac{1}{2}\int_0^1 \int_0^1 \int_x^y f(x)f(y)f(z) dzdydx$$
Any pointers would be appreciated. I have a feeling the $\int_0^x$ and $\int_x^y$ and $\int_y^1$ for z will be equal giving the factor of 3?

I knew there is a simpler way:
$$\eqalign{\left(\int_0^1 f(x)\>dx\right)^3&=\int_{[0,1]^3}f(x)f(y)f(z)\>{\rm d}(x,y,z)\cr &=6\int_{0\leq x\leq z\leq y\leq1}f(x)f(y)f(z)\>{\rm d}(x,y,z)\cr &=6\int_0^1\int_x^1\int_x^y f(x)f(y)f(z)\>dz\,dy\,dx\ .\cr}$$
