Uppgift 5

I polära koordinater beskrivs området av
$$ 1\le r\le3,\qquad\frac\pi4\le\theta\le\frac{5\pi}{4}. $$
Alltså
$$ \begin{aligned} \iint_D\frac{y^2}{(x^2+y^2)^2}\,dx\,dy &=\int_1^3\int_{\pi/4}^{5\pi/4}\frac{(r\sin\theta)^2}{r^4}\,r\,d\theta\,dr\\ &=\int_1^3\frac{dr}{r}\int_{\pi/4}^{5\pi/4}\sin^2\theta\,d\theta\\ &=\ln3\left[\frac\theta2-\frac{\sin(2\theta)}4\right]_{\pi/4}^{5\pi/4}\\ &=\frac{\pi\ln3}{2}. \end{aligned} $$