Evaluate $$ \iint _R \frac{xy^2}{(4x^2+y^2)^2}\,dA, $$where $R$ is the finite region enclosed by the parabola $y=x^2$ and the line $y=2x$.
So far I have figured out the $R$ (if I am correct) as $0\le x\le 2$ and $x^2\le y\le 2x$ but I'm stuck on how to calculate the double integral. Any help would be appreciated!
Most easy is to integrate firstly by one variable and then by another: $$\int\limits_{0}^{2}\,dx\int\limits_{x^2}^{2x}\frac{xy^2}{(4x^2+y^2)^2}\,dy= \int\limits_{0}^{4}\,dy\int\limits_{\frac{y}{2}}^{\sqrt{y}}\frac{xy^2}{(4x^2+y^2)^2}\,dx$$