Triple integral - incorrect boundaries? I'm trying to find the result of
$$   
\iiint\limits_{U} e^{x+y+z} \,dz\,dy\,dx
$$ 
where
$$
U : x \le 0,  
-x\le y \le 1,
0\le z \le -x
$$
I got the boundaries: $$ -1 \le x \le 0,\\  
0\le y \le 1,\\
0\le z \le 1
$$
And
$$   
\int\limits_{-1}^0\int\limits_{0}^{1}\int\limits_{0}^1 e^{x+y+z} \,dz\,dy\,dx = (e-1) \int\limits_{-1}^0\int\limits_{0}^{1} e^{x+y} \,dy\,dx= (e-1)^2 \int\limits_{-1}^0 e^{x} \,dx = \frac{(e-1)^3}{e} 
$$ 
But that answer is incorrect. Did I choose the wrong boundaries?
 A: Let $x\mapsto -x$ so that we want to integrate
$$\iiint_{U'}e^{-x+y+z}dz\,dy\,dx$$
where $U'$ is the region defined by $x\geq0,x\leq y\leq 1,0\leq z\leq x$. This is of course equivalent to $$0\leq z\leq x\leq y\leq 1.$$ For fixed $x$ and $y$, the value of $z$ can range from $0$ to $x$. For fixed $x$, the value of $y$ can range from $x$ to $1$. Finally, $x$ ranges from $0$ to $1$. So the integral we should get is
$$\int_0^1\int_x^1\int_0^xe^{-x+y+z}dz\,dy\,dx=3-e.$$
Where you went wrong was in finding the limits of integration. In particular, the limits of integration shouldn't be constant values which denote the maximum and minimum attainable values of each variable across all possible values of the other variables as you have done, but the values of the outer variables need to be considered as given fixed values. So the question you ask when you try to find the boundaries of integration for $z$ shouldn't be, "what's the largest and smallest possible value of $z$?" It should instead be "what's the largest and smallest possible value of $z$, given that $0\leq z\leq x$ for fixed $x$?"
