Compute an improper integral. Suppose $A = [0,\infty) \times [0, \infty) $. Let $f(x,y) = (x+y)e^{-x-y} $. How can I find $ \int_A f $? I know since $f$ is continuous on $A$, then $\int _A f $ exists, Do I need to evaluate
$$ \int_{0}^{\infty} \int_{0}^{\infty} (x+y)e^{-x-y}\ dx\ dy\ ??$$

HINT :
Rewrite:
$$ \int_{0}^{\infty} \int_{0}^{\infty} (x+y)e^{-x-y} dx dy =\int_{0}^{\infty} e^{-y} \int_{0}^{\infty} (x+y)e^{-x} dx dy\tag1
$$
Note that
$$
\int_0^\infty z^n e^{-z}\ dz=(n+1)!\tag2
$$
Use $(2)$ to evaluate $(1)$.
