Prove that the function $$ f(x, y) = \frac{(x - y) \sin(xy)}{(x^2 + y^2)^2}$$ isintegrable on $[-1, 1] \times [-1, 1]$.
My idea goes like this
Let $x=r\cos\theta, y=r\sin\theta$
We have $|f|\le {{2r^3}\over {r^4}}=2/r$ hence $\int\int |f(r,t)| rdrdt\le 2\int dr\cdot \int dt$.
I'm confused that how can I know the starting and the ending points of these integrals after I change the variables $x,y$ into polar coordinates. Could someone show me how can I determine these? Thanks in advance!