$r$

$r$

$0$

$0$

$1$

$1$

$0$

$0$

$\frac{\mathrm{\pi }}{2}$

$\frac{1}{2}\mathrm{\π}$

$\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{CenterOfMass}\right]\left(\, r\=..\,\mathrm{\theta }\=..\,\mathrm{coordinates}\=\mathrm{polar}\left[r\, \mathrm{\theta }\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^{\frac{1}{2}\mathrm{\π}}{\∫}_0^1\cos \left(\mathrm{\θ}\right)r^3\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{\frac{1}{2}\mathrm{\π}}{\∫}_0^1{r}^2\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{\frac{1}{2}\mathrm{\π}}{\∫}_0^1\sin \left(\mathrm{\θ}\right)r^3\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{\frac{1}{2}\mathrm{\π}}{\∫}_0^1{r}^2\ⅆr\ⅆ\mathrm{\θ}}$

$\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{CenterOfMass}\right]\left(\,r\=..\,\mathrm{\theta }\=..\,\mathrm{coordinates}\=\mathrm{polar}\left[r\, \mathrm{\theta }\right]\right)$

$\frac{3}{2}\frac{\sqrt{2}}{\mathrm{\π}}\,\frac{1}{4}\mathrm{\π}$

$\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{CenterOfMass}\right]\left(\,r\=..\,\mathrm{\theta }\=..\, \mathrm{coordinates}\=\mathrm{polar}\left[r\, \mathrm{\theta }\right]\,\mathrm{output}\=\mathrm{plot}\,\mathrm{caption}\=''''\right)$