This is rather a puzzle than a question, that I'm unable to solve. Consider the problem of computing the following definite integration $$\iiint_A (x+yz)\,dx\,dy\,dz \tag{1}$$ where the region $A$ is given by the set of equations $$0<x+y+z<1$$ $$0<y+z<1$$ $$0<z<1$$ We can easily rewrite $(1)$ as $$\int_{0}^{1}\int_{-z}^{1-z}\int_{-y-z}^{1-y-z}(x+yz)\,dx\,dy\,dz \tag{2}$$ and compute it as $-\frac{1}{12}$.
The puzzle is: Is it possible to exchange integrals in $(2)$ to write $(1)$ in the following form: $$\int_{a}^{b}\int_{g_1(y)}^{g_2(y)}\int_{f_1(x,y)}^{f_2(x,y)}(x+yz)\,dz\,dx\,dy \tag{3}$$ If yes, then what will be the limits of the integrals$?$
I'm almost sure that we shall have $f_1(x,y)=-x-y$ and $f_2(x,y)=1-x-y$ but having trouble figuring out the rest of the puzzle. If anyone wants to do trial and error, here's the link for wolfram alpha's triple integral calculator : http://www.wolframalpha.com/widgets/view.jsp?id=bf8679a50a63113b582ed22679363a4
Thank you.