I need to calculate
$$\lim_{n\to\infty}\int_A{(|x|(1-|x|))^{(1/n)}\arctan(ny)\over{1+x^2+y^2}} \,dx\,dy,$$
where
$$A = \left\lbrace w\in \Bbb{R^2}: |w|\le \sin(3\cdot \angle(w,e_1)) \right\rbrace,$$
where $e_1=(0,1)\in \Bbb{R^2}$. I know that I have to use Lebesgue's theorem and I know that
$$\pm\pi\over2(1+x^2+y^2)$$ depending on $\text{sgn}(y)$. But I got problem with understanding set $A$.
I think you can use $$ \angle (w,e_1)=\arccos\frac{w\cdot e_1}{|w|\cdot |e_1|} $$ and $$ \sin(3t)=\sin t(4\cos^2t-1) $$ to get a not too complicated inequality for $w=(x,y)$. Since you are at the level of knowing/learning Lebesgue theory, I believe that you can now handle the inequality in more detail.