I am having a problem with a question asked in GATE $2022$ exam.The question is as follows:
Let $f(x,y)=\begin{cases} \frac{x^2-y^2}{(x^2+y^2)^2},\text{ if } (x,y)\neq (0,0)\\ 0,\text{ if } (x,y)=(0,0)\end{cases}$ is defined on $D=[-1,1]\times[-1,1]$ then are the integrals $\iint_D|f(x,y)|dxdy$ and $\iint_D |f(x,y)|^{1/2}dxdy$ finite?
I cannot use iterated integrals because I don't know whether iterated integrals equal to this integral.I don't know any other way to check that the integrals are finite.Can someone help me by suggesting a proper approach to such problems?
Using $|x^2-y^2|\leq x^2 + y^2$ we can deduce that $$ I=\int\int_D |f(x,y)|^{1/2}dxdy\leq\int\int_D \frac{1}{|x^2+y^2|^{1/2}}. $$ Now we can integrate over a bigger disk, for instance $D\subseteq B = B_{10}(0)$, i.e. a disk located at 0 with radius 10, and using polar coordinates we get then: $$ I\leq\int_0^{10}\int_0^{2\pi}\frac{1}{r} d\varphi\text{ }rdr<\infty. $$
On the other hand using that $C=B_1(0)\subseteq D$ we can once again use polar coordinates to get
$$ J=\int\int_D|f(x,y)|dxdy\geq\int_0^{2\pi}\int_0^1\frac{|\cos^2(\varphi)-\sin^2(\varphi)|r^2}{r^4}rdrd\varphi. $$
Using Fubini (the non-negative version allows also to handle possibly infinite functions) we get that
$$ J\geq a\int_0^1\frac{1}{r}dr=\infty, $$
where $a=\int_0^{2\pi}|\cos^2(\varphi)-\sin^2(\varphi)|d\varphi>0$.