Uppgift 3

I polära koordinater $\begin{cases}x=r\cos(\theta)\\y=r\sin(\theta)\end{cases}$ motsvaras området $D$ av området $E=\{(r,\theta)\in\mathbb R^2\mid1\le r\le2,\ -\pi/2\le\theta\le\pi/4\}$ i $r\theta$-planet. Vi får
$$\begin{aligned} \iint_D\frac{x^2-y^2}{x^2+y^2}\,dx\,dy &=\iint_E\frac{r^2\cos^2(\theta)-r^2\sin^2(\theta)}{r^2}r\,dr\,d\theta =\iint_Er(\cos^2(\theta)-\sin^2(\theta))\,dr\,d\theta\\ &=\int_{-\pi/2}^{\pi/4}\cos(2\theta)\,d\theta\int_1^2r\,dr =\left[\frac{\sin(2\theta)}2\right]_{\theta=-\pi/2}^{\theta=\pi/4}\left[\frac{r^2}2\right]_{r=1}^{r=2} =\frac{1-0}2\cdot\frac{4-1}2=\frac34. \end{aligned}$$