$$$z$

$z$

$$$r$

$r$

$1$

$1$

$0$

$0$

$1$

$1$

$0$

$0$

$\frac{\mathrm{\pi }}{3}$

$\frac{1}{3}\mathrm{\π}$

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

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

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

$\frac{8}{5\mathrm{\π}}\,\frac{1}{6}\mathrm{\π}\,\frac{4}{5}$