I want to check whether $\int_{A} ze^{x+y} d(x,y,z)$ for $A=\{(x,y,z)\in\Bbb R^{3}|y<z^{2}<x<1 \\ $ does exist and if so I want to find its value.
I wanted to look at the iterated Integral of the absolute value of the function and obtain the result via Fubini-Tonelli. But I have problems how to interpret $A$ as boundaries for the iterated integrals. As I get different results depending on how I choose the boundaries of $x,y,z$. E.g
$\int_{0}^{1} \int_{0}^{x}\int_{\sqrt{y}}^{\sqrt{x}} |z| e^{x+y}\,dzdydx = \frac{1}{4}(-1-2e+e^{2}) \\$
$\int_{0}^{1} \int_{-\sqrt{x}}^{\sqrt{x}}\int_{-\infty}^{z^{2}} |z| e^{x+y}\,dydzdx = \frac {e^{2}}{2}-e+\frac {1}{2} \\$
$\int_{0}^{1} \int_{-\infty}^{0}\int_{-\sqrt{x}}^{\sqrt{x}} |z| e^{x+y}\,dzdydx = 1 \\$
Which one would be the correct approach?
EDIT:
The second one is correct. For the first, you first need to double what you have (as we have $\sqrt y\le |z|\le \sqrt x$), but then you are missing a term (which is in fact your third integral—although I hadn't noticed that before). When we do the appropriate sum, we get the same answer as the second.
This is now consistent with Fubini-Tonelli: If one of the integrals exists absolutely, then they should be equal.