How to construct a function of two variable non integrable on a stripe

Question

I'm trying to construct a function $f=f(x,y)$ of two variables defined on the stripe $A=\{(x,y)\in\mathbb{R^2}, \ |y|<a\}$ such that $f\in C^2(A)$ and

$\int_Bf(x,y)dx dy=\infty$ , where $B=\{(x,y)\in\mathbb{R^2} \ |y|<b\}$, $b<a$.

How can I do? Thanks for the help!