Application of Fubini's theorem, lebesgue integral with product measure

Question

Show that for $f(x,y)=xy/(x^2+y^2)^2 $ for $ x,y \neq (0,0) $ and $ f(0,0)=0$ , the iterated integrals $\int_{-1}^1\int_{-1}^1fdxdy $ and $\int_{-1}^1\int_{-1}^1fdydx$ coincide but that the double integral $ \int_{(-1, 1)^2} f d\mu$ does not exist.

Answer 1

The double integral does not exist: using the polar coordinates, $$ f(r,\theta) = \frac {A(\theta)}{r^2} $$ and $$ \int_0^1 \frac {dr}{r^2} = \infty $$

The nested integral exist and coincide: as $f(x,y) = f(y,x)$ I just have to prove the existence of the integral. For $y>0$ fixed, $|f(x,y)| = \frac {|y|}2 \frac {2|x|}{(x^2 + y^2)^2}$ and $$ \int_0^1 \frac {2x}{(x^2 + y^2)^2} dx = \int_0^1 \frac {dU}{(U + y^2)^2} < \infty $$ and the inner integral has value 0, because $f(-x,y) =- f(x, y)$ !