Question: For $D=[0,+\infty)×[0,+\infty)$ and $f(x,y)=(1+x+y)^{-1}$, why does the integral over $D$ of $f(x,y)$ not converge? 

It's not like $\rho/(1+\rho|\sin\theta+\cos\theta|)$ which to infinite is like $1/(|\sin\theta+\cos\theta|)$ and, close to $0$ is like $0$?

Answer: 
Consider 

$$D'=[0,+\infty)\times [0,1]$$

and note that

$$\int\_D \frac1{1+x+y} \,dx\,dy \ge \int\_{D'} \frac1{1+x+y} \,dx\,dy\ge \int\_{D'} \frac1{2+x} \, dx \, dy$$
Intuitively, the area of the domain grows like $xy$, while the integrand decreases like $\dfrac1{1+x+y}$, which is insufficient to compensate.