Calculate Double integral $$\iint_D (x^2 + y^2 ) dxdy$$ where: $$D=\{(x,y)\in\mathbb{R}^2 : x\le x^2+y^2\le2x, -x\le y \le x \}$$ What i did? I tried to use polar coordinates and i got this ==> $\sqrt x\le r \le \sqrt(2x)$
Can you please help me with the limit of integration if i change this to polar coordinates. Thank you
You haven't fully used polar coordinates. Recall the transformation is given by $(x,y) \mapsto (r\cos \theta, r \sin \theta)$. From $D$ we then get the bounds: $r\cos \theta \leq r^2 \leq 2 r\cos \theta$, $-r \cos \theta \leq r\sin \theta \leq r \cos \theta$. The first one of course reduces to $\cos \theta \leq r \leq 2\cos \theta$ while if you divide the second one by $r\cos\theta$ you can get $\arctan(-1) \leq \theta \leq \arctan(1)$. From there you get an integral that is manageable to evaluate.