Change of Variables & the Jacobian · Superchallenge problem

Let $R$ be the region of points $(x, y)$ with $x, y > 0$ satisfying $2x \le x^2 + y^2 \le 4x$ and $2y \le x^2 + y^2 \le 6y$ (it is bounded by four circles through the origin). Evaluate $\displaystyle\iint_R \dfrac{dA}{(x^2 + y^2)^2}$.