${\int }_0^4{\int }_0^3{\int }_0^{{\sqrt{4-z}}_{}}1 \mathrm{dx} \mathrm{dy} \mathrm{dz}\=16$ 

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Loading Student:-MultivariateCalculus

Access the MultiInt command via the Context Panel

$1$$\stackrel{\text{MultiInt}}{\to }$${\∫}_0^4{\∫}_0^3{\∫}_0^{\sqrt{4-z}}1\ⅆx\ⅆy\ⅆz$$\=$$16$

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 3-D

Table 7.3.5(a)   Solution by visualization task template

${\int }_0^4{\int }_0^3{\int }_0^{{\sqrt{4-z}}_{}}1 \mathit{\ⅆ}x \mathit{\ⅆ}y \mathit{\ⅆ}z$ = $16$$$

Table 7.3.5(b)   Solution via iterated triple-integral template in the Calculus palette

Top-level: Int and int commands

$\mathrm{Int}\left(1\,\left[x\=0..\sqrt{4-z}\,y\=0..3\,z\=0..4\right]\right)\=\mathrm{int}\left(1\,\left[x\=0..\sqrt{4-z}\,y\=0..3\,z\=0..4\right]\right)$

${\∫}_0^4{\∫}_0^3{\∫}_0^{\sqrt{4-z}}1\ⅆx\ⅆy\ⅆz\=16$

The MultiInt command in the Student MultivariateCalculus package

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

$\mathrm{MultiInt}\left(1\,x\=0..\sqrt{4-z}\,y\=0..3\,z\=0..4\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^4{\∫}_0^3{\∫}_0^{\sqrt{4-z}}1\ⅆx\ⅆy\ⅆz$

$\mathrm{MultiInt}\left(1\,x\=0..\sqrt{4-z}\,y\=0..3\,z\=0..4\right)$ = $16$$$

$\mathrm{MultiInt}\left(1\,x\=0..\sqrt{4-z}\,y\=0..3\,z\=0..4\,\mathrm{output}\=\mathrm{steps}\right)$

$16$

The MultiInt command with a pre-defined domain option

$\mathrm{MultiInt}\left(1\,\left[z\,y\,x\right]\=\mathrm{Region}\left(0..4\,0..3\,0..\sqrt{4-z}\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^4{\∫}_0^3{\∫}_0^{\sqrt{4-z}}1\ⅆx\ⅆy\ⅆz$

$\mathrm{MultiInt}\left(1\,\left[z\,y\,x\right]\=\mathrm{Region}\left(0..4\,0..3\,0..\sqrt{4-z}\right)\right)$ = $16$$$