Not integrable although iterated integrals are equal

Question

How can I show that the function

$$f=\begin{cases} 0 & (x,y)=(0,0)\\\frac{xy}{(x^2+y^2)^2} & \mbox{else}\end{cases}$$ is not Lebesgue-integrable, although the iterated integrals exist and are equal: $$\int_{-1}^{1}\int_{-1}^{1}f(x,y)dydx=\int_{-1}^{1}\int_{-1}^{1}f(x,y)dxdy?$$

Answer 1

Hint. As $(x,y) \to (0,0)$, using polar coordinates we have, $$ f(x,y) \sim \frac{\sin \theta\cos \theta}{r^2} $$ which is not integrable as $r \to 0$.