Let $D$ be any Jordan domain in $\mathbb{R}^2$, containing origin in its interior,
whose boundary $\partial D$ has the form $r = f(\theta)$ in polar coordinates where
$f \in C[0,2\pi]$.
Consider following integral as a functional of $D$:
$$\mathcal{I}_D \stackrel{def}{=} \int_D \phi(x,y) dx dy
\quad\text{ where }\quad\phi(x,y) = \frac{\sin(x^2+y^2)}{x^2+y^2}
$$
Since the origin is a removable singularity for $\phi(x,y)$, as long as $D$ is
of finite extent, there isn't any issue about integrability or change of variable. We have
$$\mathcal{I}_D = \int_0^{2\pi} \int_0^{f(\theta)}\frac{\sin(r^2)}{r^2} rdr d\theta
= \frac12\int_0^{2\pi} \left[\int_0^{f(\theta)^2}\frac{\sin t}{t} dt \right] d\theta
$$
For any non-increasing, non-negative function $g$ on $(0,\infty)$. Using integration by part (the RS version), one can show that
$$\left|\int_a^b g(x) \sin(x) dx \right| \le 2 g(a)\quad\text{ for }\quad 0 < a < b < \infty$$
For any $R > 0$ where $B(0,R) \subset D$. By setting $g(x)$ to $1/x$, above inequality leads to following estimate for $\mathcal{I}_D$.
$$\left| \mathcal{I}_D - \mathcal{I}_{B(0,R)} \right|
= \frac12 \left| \int_0^{2\pi} \left[\int_{R^2}^{f(\theta)^2}\frac{\sin t}{t} dt \right] d\theta \right|
\le \frac12 \int_0^{2\pi} \left|\int_{R^2}^{f(\theta)^2}\frac{\sin t}{t} dt\right|
d\theta \le \frac{2\pi}{R^2}
$$
For any fixed $Y$, the integrand $\phi(x,y)$ is Lebesgue integrable over $(-\infty,\infty)\times [-Y,Y]$. Double integral of the form below
is well defined. With help of DCT, one can evaluate it as a limit
$$\int_{-Y}^Y \int_{-\infty}^{\infty}\phi(x,y) dxdy =
\lim_{X\to\infty}\int_{-Y}^Y \int_{-X}^X \phi(x,y) dxdy
= \lim_{X\to\infty}\mathcal{I}_{[-X,X]\times[-Y,Y]}$$
We will combine this with above estimation. By setting $R = Y$ and $[-X,X] \times [-Y,Y]$ taking the role of $D$, one get
$$\left|\int_{-Y}^{Y} \int_{-\infty}^{\infty}\phi(x,y) dxdy - \mathcal{I}_{B(0,Y)}\right|
\le
\limsup_{X\to\infty}\left|\int_{-Y}^{Y} \int_{-X}^{X}\phi(x,y) dxdy - \mathcal{I}_{B(0,Y)}\right|
\le \frac{2\pi}{Y^2}$$
Since following two limits exist,
$$\lim_{Y\to\infty} \mathcal{I}_{B(0,Y)} = \lim_{Y\to\infty} \pi\int_0^{Y^2}\frac{\sin t}{t}dt = \pi\int_0^\infty \frac{\sin t}{t} dt = \frac{\pi^2}{2}
\quad\text{ and }\quad
\lim_{Y\to\infty}\frac{2\pi}{Y^2} = 0$$
By squeezing, the double integral at hand exists as an improper integral!
$$\int_{-\infty}^\infty \int_{-\infty}^\infty \phi(x,y) dxdy
\stackrel{def}{=} \lim_{Y\to\infty} \int_{-Y}^Y \int_{-\infty}^\infty \phi(x,y) dxdy = \lim_{Y\to\infty} \mathcal{I}_{B(0,Y)} = \frac{\pi^2}{2}$$
