Uppgift 4

(a) En skiss ger
$$\int_1^2\int_{y^2}^4 f(x,y)\,dx\,dy =\int_1^4\int_1^{\sqrt{x}} f(x,y)\,dy\,dx.$$

(b) Med polära koordinater $x=r\cos\theta$, $y=r\sin\theta$, där $1\le r\le2$ och $\pi/4\le\theta\le\pi/2$ har vi
$$\begin{aligned} \iint_Dxy e^{(x^2+y^2)^2}\,dx\,dy &=\int_1^2\int_{\pi/4}^{\pi/2}\sin\theta\cos\theta\,r^3e^{r^4}\,d\theta\,dr\\\\ &=\frac14[e^{r^4}]_1^2\frac12[\sin^2\theta]_{\pi/4}^{\pi/2} =\frac1{8}(e^{16}-e)\left(1-\frac12\right) =\frac1{16}e(e^{15}-1). \end{aligned}$$