${\int }_{x\=a}^{x\=b}{\int }_{y\=c}^{y\=d}{\int }_{z\=e}^{z\=f}g\left(x\,y\,z\right) \mathrm{dz} \mathrm{dy} \mathrm{dx}$ $$

${\int }_{y\=c}^{y\=d}{\int }_{x\=a}^{x\=b}{\int }_{z\=e}^{z\=f}g\left(x\,y\,z\right) \mathrm{dz} \mathrm{dx} \mathrm{dy}$ $$

${\int }_{z\=e}^{z\=f}{\int }_{x\=a}^{x\=b}{\int }_{y\=c}^{y\=d}g\left(x\,y\,z\right) \mathrm{dy} \mathrm{dx} \mathrm{dz}$ $$

${\int }_{x\=a}^{x\=b}{\int }_{z\=e}^{z\=f}{\int }_{y\=c}^{y\=d}g\left(x\,y\,z\right) \mathrm{dy} \mathrm{dz} \mathrm{dx}$ $$

${\int }_{z\=e}^{z\=f}{\int }_{y\=c}^{y\=d}{\int }_{x\=a}^{x\=b}g\left(x\,y\,z\right) \mathrm{dx} \mathrm{dy} \mathrm{dz}$ $$

${\int }_{y\=c}^{y\=d}{\int }_{z\=e}^{z\=f}{\int }_{x\=a}^{x\=b}g\left(x\,y\,z\right) \mathrm{dx} \mathrm{dz} \mathrm{dy}$ $$

Table 7.2.1   In Cartesian coordinates, the six iterations of a triple integral

${\int }_{{x_1}_{}}^{{x_2}_{}}{\int }_{{y_1}_{}}^{{y_2}_{}}{\int }_{{z_1}_{}}^{{z_2}_{}}f\ⅆz\ⅆy\ⅆx$, the iterated triple-integral template in the Calculus palette

The MultiInt command in the Student MultivariateCalculus package

The task template at

Tools≻Tasks≻Browse: Calculus - Vector≻Integration≻Multiple Integration≻3-D≻Over a Cube

The task template at Tools≻Tasks≻Browse:

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

The Parallelepiped option in the modified int command of the Student VectorCalculus package

The Int and int commands at top-level

Table 7.2.2   Maple tools for iterating a triple integral over a "box"

Example 7.2.1

$f\left(x\,y\,z\right)\=x y z$; $R$ is defined by the inequalities $0\le x\,y\,z\le 1$.

See Example 7.1.1.

Example 7.2.2

$f\left(x\,y\,z\right)\=1\+2 x\+3 y\+5 z$; $R$ is defined by the inequalities $1\le x\,y\,z\le 2$.

See Example 7.1.2.

Example 7.2.3

$f\left(x\,y\,z\right)\=1\+2 x^2\+3 y^2\+4 z^2$; $R\=\{\left(x\,y\,z\right)\| x\in \left[1\,2\right]\,y\in \left[-1\,3\right]\,z\in \left[2\,4\right]\}$.

See Example 7.1.3.