$z$

$z$

$r$

$r$

$1$

$1$

$0$

$0$

$1$

$1$

$0$

$0$

$2 \mathrm{\pi }$

$2\mathrm{\π}$

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

${\∫}_0^{2\mathrm{\π}}{\∫}_0^1{\∫}_r^{1}zr\ⅆz\ⅆr\ⅆ\mathrm{\θ}$

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

$\frac{1}{4}\mathrm{\π}$

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

$\begin{array}{ccc}\multicolumn{3}{c}{{\int }_0^{2\mathrm{\pi }}{\int }_0^1{\int }_r^{1}zr\ⅆz\ⅆr\ⅆ\mathrm{\theta }}\\ & \text{=}& {\int }_0^{2\mathrm{\pi }}{\int }_0^1\left(\genfrac{}{}{0ex}{}{\frac{z^{2}r}{2}}{\phantom{z=r..1}}|\genfrac{}{}{0ex}{}{\phantom{\frac{z^{2}r}{2}}}{z=r..1}\right)\ⅆr\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& {\int }_0^{2\mathrm{\pi }}{\int }_0^1\frac{r\left(1-r^2\right)}{2}\ⅆr\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& {\int }_0^{2\mathrm{\pi }}\left(\genfrac{}{}{0ex}{}{\left(-\frac{1}{8}{r}^4+\frac{1}{4}{r}^2\right)}{\phantom{r=0..1}}|\genfrac{}{}{0ex}{}{\phantom{\left(-\frac{1}{8}{r}^4+\frac{1}{4}{r}^2\right)}}{r=0..1}\right)\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& {\int }_0^{2\mathrm{\pi }}\frac{1}{8}\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& \genfrac{}{}{0ex}{}{\frac{\mathrm{\theta }}{8}}{\phantom{\mathrm{\theta }=0..2\mathrm{\pi }}}|\genfrac{}{}{0ex}{}{\phantom{\frac{\mathrm{\theta }}{8}}}{\mathrm{\theta }=0..2\mathrm{\pi }}\hfill \end{array}$

$\frac{1}{4}\mathrm{\π}$