Let $f(x,y) = \dfrac{1}{(y+1)^2}$ and let $A$ and $B$ be the open subsets
$A = \{(x,y)\,|\,x > 0 \text{ and } x < y < 2x\}$
$B = \{(x,y)\,|\,x > 0 \text{ and } x^2 < y < 2x^2\}$
of $\mathbb{R}^2$.
How to show that the $\int_A f$ does not exist but the $\int_B f$ does and find its value?