$Z\left(r\,\mathrm{\θ}\right)\=\sqrt{2- r^2}$ 

   while the lower surface is $z\left(r\,\mathrm{\θ}\right)\= r^2$.

${\int }_0^{2 \mathrm{\π}}{\int }_0^1{\int }_{ r^2}^{\sqrt{2- r^2}}r \mathrm{dz} \mathrm{dr} d\mathrm{\θ}\=\frac{\mathrm{\π}}{6}\left(8\sqrt{2}-7\right)\doteq 2.26$ 

$r^2\+z^2\=2\,z\=r^2$$\stackrel{\text{solve}}{\to }$$\left\{r\=1\,z\=1\right\}\,\left\{r\=-1\,z\=1\right\}\,\left\{r\=\mathrm{I}\sqrt{2}\,z\=-2\right\}\,\left\{r\=-\mathrm{I}\sqrt{2}\,z\=-2\right\}$

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cylindrical

Table 7.4.5(a)   Task template for iterating a triple integral in cylindrical coordinates

${\int }_0^{2 \mathrm{\π}}{\int }_0^1{\int }_{ r^2}^{\sqrt{2- r^2}}r \mathit{\ⅆ}z \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{4}{3}\sqrt{2}\mathrm{\π}-\frac{7}{6}\mathrm{\π}$$$

Table 7.4.5(b)   Solution from first principles

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

Table 7.4.5(c)   Solution by task template that implements the MultiInt command

Initialize

Loading Student:-MultivariateCalculus

Access the MultiInt command via the Context Panel

$1$$\stackrel{\text{MultiInt}}{\to }$${\∫}_0^{2\mathrm{\π}}{\∫}_0^1{\∫}_{r^2}^{\sqrt{-r^2\+2}}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}$$\=$$\frac{4}{3}\sqrt{2}\mathrm{\π}-\frac{7}{6}\mathrm{\π}$$\stackrel{\text{at 5 digits}}{\to }$$2.2584$

Initialize

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

$\mathrm{solve}\left(\left\{r^2\+z^2\=2\,z\=r^2\right\}\,\mathrm{explicit}\right)$

$\left\{r\=1\,z\=1\right\}\,\left\{r\=-1\,z\=1\right\}\,\left\{r\=\mathrm{I}\sqrt{2}\,z\=-2\right\}\,\left\{r\=-\mathrm{I}\sqrt{2}\,z\=-2\right\}$

Top-level solution using the Int and int commands

$\mathrm{Int}\left(r\,\left[z\= r^2..\sqrt{2- r^2}\,r\=0..1\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)\=\mathrm{int}\left(r\,\left[z\= r^2..\sqrt{2- r^2}\,r\=0..1\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^1{\∫}_{r^2}^{\sqrt{-r^2\+2}}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}\=2\left(\frac{2}{3}\sqrt{2}-\frac{7}{12}\right)\mathrm{\π}$

Solution via the MultiInt command from the Student MultivariateCalculus package

$$\begin{gathered} \mathrm{MultiInt}\left(1\,z\= r^2..\sqrt{2- r^2}\,r\=0..1\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\,\mathrm{output}\=\mathrm{integral}\right)\= \\ \mathrm{MultiInt}\left(1\,z\= r^2..\sqrt{2- r^2}\,r\=0..1\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^1{\∫}_{r^2}^{\sqrt{-r^2\+2}}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}\=\frac{4}{3}\sqrt{2}\mathrm{\π}-\frac{7}{6}\mathrm{\π}$

$\mathrm{MultiInt}\left(1\,z\= r^2..\sqrt{2- r^2}\,r\=0..1\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\,\mathrm{output}\=\mathrm{steps}\right)$

$\frac{4}{3}\sqrt{2}\mathrm{\π}-\frac{7}{6}\mathrm{\π}$