For functions $f:\mathbb{R}\to \mathbb{R}$ the definition of a convergent improper integral is straightforward: the integral $\int_\mathbb{R} f(x)dx$ converges iff $$\lim_{(a,b)\to (-\infty, \infty)}\int_a^b f(x) dx $$ exists. However, already in dimension 2 things get trickier. For example, if $f(x,y)=\sin (x^2+y^2)$ and we take $$\int_{-a\leq x,y\leq a}f(x,y) dxdy$$ the limit as $a\to\infty$ exists and is equal $\pi$.
On the other hand, if we replace the square $\{(x,y)\mid -a\leq x,y\leq a\}$ by the disk of radius $a$ centered at the origin as the integration domain, the limit $$\lim_{a\to+\infty}\int_{x^2+y^2\leq a^2}f(x,y) dxdy$$ will no longer exist. In fact, the value will oscillate around $\pi$. Also, it is clear that for any $A\in\mathbb{R}$ one can find a continuous family $\mathbb{R}\ni t\mapsto X_t$ of compact connected polygons in $\mathbb{R}^2$ such that $$\lim_{t\to\infty} \int_{X_t}f(x,y) dxdy=A.$$
Here is my first question: is it possible to find a continous family $(X_t)$ of compact and convex polygons in $\mathbb{R}^2$ such that $$\lim_{t\to\infty} \int_{X_t}f(x,y) dxdy$$ would exist but would be $\neq \pi$? If not, then is there a smooth function $g:\mathbb{R}^2\to \mathbb{R}$ and two continous families $(X_t),(Y_t)$ as above (i.e., $X_t,Y_t$ are compact and convex), such that $$\lim_{t\to\infty} \int_{X_t}g(x,y) dxdy,\lim_{t\to\infty} \int_{Y_t}g(x,y) dxdy$$ would both exist but would be different?
More generally, is there a standard way to regularize integrals such as $\int_{\mathbb{R}^2}f(x,y) dxdy$? One possible way would be to replace $f(x,y)$ by $f(x,y)h(tx,ty)$ where $h:\mathbb{R}^2\to\mathbb{R}$ is absolutely integrable and then take the limit as $t\to 0$. But it is not quite clear whether this procedure will be independent of $h$, even if one takes $h=e^{-q(x,y)}$ with $q$ a positive definite quadratic polynomial (not necessarily homogeneous).
The answer to the modified version of question 1 is yes. Let $g_0:\mathbb{R}\to\mathbb{R}$ be given by $x\mapsto x/(1+x^4)$ (basically we need an odd function which would be take positive values on the positive half-axis and whose integral would converge absolutely). Set $g(x,y)=g_0(x)$. Let $A$ be a real number $\geq 0$. For $t\geq 0$ set $\epsilon(t)=\int_t^{2t}g_0(t) dt$ and take $$X^A_t=\{(x,y)\in\mathbb{R}^2\mid -t\leq x\leq 2t,\frac{-A}{2\epsilon(t)}\leq y\leq\frac{A}{2\epsilon(t)}\}.$$ Then $\int_{X^A_t}g(x,y)dxdy =A$ for all $t$. Similarly, if one takes $$Y^A_t=\{(x,y)\in\mathbb{R}^2\mid -2t\leq x\leq t,\frac{-A}{2\epsilon(t)}\leq y\leq\frac{A}{2\epsilon(t)}\},$$ then $\int_{Y^A_t}g(x,y)dxdy =-A$ for all $t$