${\int }_{x\=1}^{x\=2}{\int }_{y\=1}^{y\=2}{\int }_{z\=1}^{z\=2}f\left(x\,y\,z\right) \mathrm{dz} \mathrm{dy} \mathrm{dx}\=16$ 

${\int }_{y\=1}^{y\=2}{\int }_{x\=1}^{x\=2}{\int }_{z\=1}^{z\=2}f\left(x\,y\,z\right) \mathrm{dz} \mathrm{dx} \mathrm{dy}\=16$ 

${\int }_{z\=1}^{z\=2}{\int }_{x\=1}^{x\=2}{\int }_{y\=1}^{y\=2}f\left(x\,y\,z\right) \mathrm{dy} \mathrm{dx} \mathrm{dz}\=16$ 

${\int }_{x\=1}^{x\=2}{\int }_{z\=1}^{z\=2}{\int }_{y\=1}^{y\=2}f\left(x\,y\,z\right) \mathrm{dy} \mathrm{dz} \mathrm{dx}\=16$ 

${\int }_{z\=1}^{z\=2}{\int }_{y\=1}^{y\=2}{\int }_{x\=1}^{x\=2}f\left(x\,y\,z\right) \mathrm{dx} \mathrm{dy} \mathrm{dz}\=16$ 

${\int }_{y\=1}^{y\=2}{\int }_{z\=1}^{z\=2}{\int }_{x\=1}^{x\=2}f\left(x\,y\,z\right) \mathrm{dx} \mathrm{dz} \mathrm{dy}\=16$ 

Table 7.2.2(a)   In Cartesian coordinates, the six iterations of a given triple integral

${\int }_1^2{\int }_1^2{\int }_1^2\left(1\+2 x\+3 y\+5 z\right) \mathrm{dz} \mathrm{dy} \mathrm{dx}$

$\={\int }_1^2{\int }_1^2{\left[z\left(1\+2 x\+3 \right)\+\frac{5 z^2}{2}\right]}_{z\=1}^{z\=2} \mathrm{dy} \mathrm{dx}$

$\={\int }_1^2{\int }_1^2\left(\left(2-1\right)\left(1\+2 x\+3 y\right)\+\frac{5}{2}\left(2^2-1^2\right)\right) \mathrm{dy} \mathrm{dx}$

$\={\int }_1^2{\int }_1^2\left(\frac{17}{2}\+2 x\+3 y\right) \mathrm{dy} \mathrm{dx}$

$\={\int }_1^2{\left[\frac{17}{2}\+2 x y\+\frac{3 y^2}{2}\right]}_{y\=1}^{y\=2} \mathrm{dx}$

$\={\int }_1^2\left(\frac{17}{2}\+2 x \left(2-1\right)\+\frac{3}{2}\left(2^2-1^2\right)\right) x \mathrm{dx}$

$\={\int }_1^2\left(13\+2 x\right) \mathrm{dx}$

$\={\left[13 x\+2\frac{x^2}{2}\right]}_{x\=1}^{x\=2}$

$\=13\left(2-1\right)\+\left(2^2-1^2\right)$

$\=16$

Table 7.2.2(b)   Stepwise evaluation of the $\mathrm{dz} \mathrm{dy} \mathrm{dx}$ iteration

Define the integrand $f$ 

$f\=1\+2 x\+3 y\+5 z$$\stackrel{\text{assign}}{\to }$

Write and evaluate all six iterations of the triple integral

${\int }_1^2{\int }_1^2{\int }_1^{2}f \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $16$

${\int }_1^2{\int }_1^2{\int }_1^{2}f \mathit{\ⅆ}z \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $16$

${\int }_1^2{\int }_1^2{\int }_1^{2}f \mathit{\ⅆ}y \mathit{\ⅆ}x \mathit{\ⅆ}z$ = $16$

${\int }_1^2{\int }_1^2{\int }_1^{2}f \mathit{\ⅆ}y \mathit{\ⅆ}z \mathit{\ⅆ}x$ = $16$

${\int }_1^2{\int }_1^2{\int }_1^{2}f \mathit{\ⅆ}x \mathit{\ⅆ}y \mathit{\ⅆ}z$ = $16$

${\int }_1^2{\int }_1^2{\int }_1^{2}f \mathit{\ⅆ}x \mathit{\ⅆ}z \mathit{\ⅆ}y$ = $16$

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Multiple Integration≻3-D≻Over a Cube

Table 7.2.2(c)   Solution by task template

Tools≻Tasks≻Browse:

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

Table 7.2.2(d)   Task template for visualizing regions of integration

Initialize

$f≔1\+2 x\+3 y\+5 z\:$

Write and evaluate all six iterations of the triple integral

$\mathrm{Int}\left(f\,\left[z\=1..2\,y\=1..2\,x\=1..2\right]\right)\=\mathrm{int}\left(f\,\left[z\=1..2\,y\=1..2\,x\=1..2\right]\right)$

${\∫}_1^2{\∫}_1^2{\∫}_1^2\left(1\+2x\+3y\+5z\right)\ⅆz\ⅆy\ⅆx\=16$

$\mathrm{Int}\left(f\,\left[z\=1..2\,x\=1..2\,y\=1..2\right]\right)\=\mathrm{int}\left(f\,\left[z\=1..2\,x\=1..2\,y\=1..2\right]\right)$

${\∫}_1^2{\∫}_1^2{\∫}_1^2\left(1\+2x\+3y\+5z\right)\ⅆz\ⅆx\ⅆy\=16$

$\mathrm{Int}\left(f\,\left[y\=1..2\,x\=1..2\,z\=1..2\right]\right)\=\mathrm{int}\left(f\,\left[y\=1..2\,x\=1..2\,z\=1..2\right]\right)$

${\∫}_1^2{\∫}_1^2{\∫}_1^2\left(1\+2x\+3y\+5z\right)\ⅆy\ⅆx\ⅆz\=16$

$\mathrm{Int}\left(f\,\left[y\=1..2\,z\=1..2\,x\=1..2\right]\right)\=\mathrm{int}\left(f\,\left[y\=1..2\,z\=1..2\,x\=1..2\right]\right)$

${\∫}_1^2{\∫}_1^2{\∫}_1^2\left(1\+2x\+3y\+5z\right)\ⅆy\ⅆz\ⅆx\=16$

$\mathrm{Int}\left(f\,\left[x\=1..2\,y\=1..2\,z\=1..2\right]\right)\=\mathrm{int}\left(f\,\left[x\=1..2\,y\=1..2\,z\=1..2\right]\right)$

${\∫}_1^2{\∫}_1^2{\∫}_1^2\left(1\+2x\+3y\+5z\right)\ⅆx\ⅆy\ⅆz\=16$

$\mathrm{Int}\left(f\,\left[x\=1..2\,z\=1..2\,y\=1..2\right]\right)\=\mathrm{int}\left(f\,\left[x\=1..2\,z\=1..2\,y\=1..2\right]\right)$

${\∫}_1^2{\∫}_1^2{\∫}_1^2\left(1\+2x\+3y\+5z\right)\ⅆx\ⅆz\ⅆy\=16$

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{MultiInt}\left(f\,z\=1..2\,y\=1..2\,x\=1..2\,\mathrm{output}\=\mathrm{steps}\right)$

$16$

Table 7.2.2(e)   Stepwise evaluation of an iterated triple-integral

$\mathrm{MultiInt}\left(f\,\left[x\,y\,z\right]\=\mathrm{Parallelepiped}\left(1..2\,1..2\,1..2\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_1^2{\∫}_1^2{\∫}_1^2\left(1\+2x\+3y\+5z\right)\ⅆx\ⅆy\ⅆz$

$\mathrm{MultiInt}\left(f\,\left[x\,y\,z\right]\=\mathrm{Parallelepiped}\left(1..2\,1..2\,1..2\right)\right)$ = $16$

Table 7.2.2(f)  The Parallelepiped option in the MultiInt command, Student MultivariateCalculus