Question

Sin(x2 + y2) da r , where r is the region in the first quadrant between the circles with center the origin and radii 2 and 5

Answer 1

Converting to polar coordinates gives

$\displaystyle\iint_R\sin(x^2+y^2)\,\mathrm dA=\int_{\theta=0}^{\theta=\pi/2}\int_{r=2}^{r=5}\sin(r^2)r\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\pi\int_{r=2}^{r=5}2r\sin(r^2)\,\mathrm dr$
$=\displaystyle\pi\int_{r^2=4}^{r^2=25}\sin(r^2)\,\mathrm d(r^2)$
$=-\cos(r^2)\bigg|_{r^2=4}^{r^2=25}$
$=\pi(\cos4-\cos25)$