I want to compute $\iiint_{\Delta}xyz(1-x-y-z)dx dydz$ over $\Delta=${$x,y,z \space | \space x,y,z>0, x+y+z<1$}
The following change of variable is suggested:
$X=x+y+z, \space XY=y+z, \space XYZ=z$
How do I find the bounds for $X$, $Y$, and $Z$ ? In what way can I always find them? It seems like no matter what I do, I always end up in a circular reasoning or something, the end result being $0<X<1$ which I already know. And it can't be the only condition, else the integral would go to infinity.
For example I could write that $0<x<1$, $0<y<1-x$ and $0<z<1-y-x$ and I know that $z=XYZ, y=XY(1-Z), x=X(1-Y)$ but no matter how I try to find inequalities, I always end up with $0<X<1$
If it matters, I found a jacobian equal to $1$ and was able to rewrite the integral into:
$\iiint_{??} (1-X)(1-Y)(1-Z)X^3 Y^2Z\space dXdYdZ$
(assuming I didn't make any mistakes in the process)
$X$ goes from $0$ to $1$, but can't find the bounds for $Y$ and $Z$ for the life of me.