${\int }_0^{2 \mathrm{\π}}{\int }_0^3{\int }_0^{9-r^2}r \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{81}{2} \mathrm{\π}$ 

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cylindrical

Table 8.1.20(a)   Task template integrating in cylindrical coordinates

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Multiple Integration≻Cylindrical

Table 8.1.20(b)   Task template implementing the MultiInt command iterating in the order $\mathrm{dz} \mathrm{dr} d\mathrm{\θ}$

Initialize

Loading Student:-MultivariateCalculus

Access the MultiInt command via the Context Panel

$1$$\stackrel{\text{MultiInt}}{\to }$${\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}$$\=$$\frac{81}{2}\mathrm{\π}$

${\int }_0^{2 \mathrm{\π}}{\int }_0^3{\int }_0^{9-r^2}r \mathit{\ⅆ}z \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{81}{2}\mathrm{\π}$$$

Table 8.1.20(c)   Integration via first principles

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$\mathrm{MultiInt}\left(1\,z\=0..9-r^2\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\θ}\,z\right]\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}$

$\mathrm{MultiInt}\left(1\,z\=0..9-r^2\,r\=0..3\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\right)$ = $\frac{81}{2}\mathrm{\π}$$$

Table 8.1.20(d)   MultiInt command iterating in cylindrical coordinates in the order $\mathrm{dz} \mathrm{dr} d\mathrm{\θ}$

$$\begin{gathered} \mathrm{Int}\left(r\,\left[z\=0..9-r^2\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)\= \\ \mathrm{int}\left(r\,\left[z\=0..9-r^2\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}\=\frac{81}{2}\mathrm{\π}$

Table 8.1.20(e)   Top-level Int and int commands