Uppgift 5

I polära koordinater ges området av $1\le r\le2$, $-\pi/4\le\theta\le3\pi/4$. Vi får
$$\begin{aligned} \iint_Dx^2y^2\,dx\,dy &=\int_1^2\int_{-\pi/4}^{3\pi/4}(r\cos\theta)^2(r\sin\theta)^2r\,d\theta\,dr\\ &=\int_1^2r^5\,dr\int_{-\pi/4}^{3\pi/4}\cos^2\theta\sin^2\theta\,d\theta \end{aligned}$$
($\cos^2\theta\sin^2\theta=\frac14\sin^22\theta=\frac18(1-\cos4\theta)$)
$$\begin{aligned} &=\left[\frac16r^6\right]_1^2\left[\frac\theta8-\frac1{32}\sin4\theta\right]_{-\pi/4}^{3\pi/4}\\ &=\left(\frac{64}6-\frac16\right)\cdot\left(\frac{3\pi}{32}+\frac\pi{32}\right)=\frac{21}2\cdot\frac\pi8=\frac{21\pi}{16}. \end{aligned}$$