Change of variable in double integral. Domain transformation.

Question

$$\iint_D(\frac{y^2 - x^2}{(x^2 + y^2)^2})dxdy $$ $$D = {(x-1)^2+(y-1)^2<1}$$ Could you please help me? I tried to apply the transformation to polar coordinates and yet I don't know how to transform the domain correctly.

Answer 1

Letting $x\to y$ and $y\to x$, one has $$ \iint_D\frac{y^2}{(x^2 + y^2)^2}dxdy=\iint_D\frac{x^2}{(x^2 + y^2)^2}dxdy $$ and heence \begin{eqnarray} \iint_D\frac{y^2 - x^2}{(x^2 + y^2)^2}dxdy=0. \end{eqnarray}