${\int }_{x\=a}^{x\=A}{\int }_{y\=y\left(x\right)}^{y\=Y\left(x\right)}{\int }_{z\=z\left(x\,y\right)}^{z\=Z\left(x\,y\right)}f\left(x\,y\,z\right) \mathrm{dz} \mathrm{dy} \mathrm{dx}$ 

${\int }_{y\=b}^{y\=B}{\int }_{x\=x\left(y\right)}^{x\=X\left(y\right)}{\int }_{z\=z\left(x\,y\right)}^{z\=Z\left(x\,y\right)}f\left(x\,y\,z\right) \mathrm{dz} \mathrm{dx} \mathrm{dy}$ 

${\int }_{z\=c}^{z\=C}{\int }_{x\=x\left(z\right)}^{x\=X\left(z\right)}{\int }_{y\=y\left(x\,z\right)}^{y\=Y\left(x\,z\right)}f\left(x\,y\,z\right) \mathrm{dy} \mathrm{dx} \mathrm{dz}$ 

${\int }_{x\=a}^{x\=A}{\int }_{z\=z\left(x\right)}^{z\=Z\left(x\right)}{\int }_{y\=y\left(x\,z\right)}^{y\=Y\left(x\,z\right)}f\left(x\,y\,z\right) \mathrm{dy} \mathrm{dz} \mathrm{dx}$ 

${\int }_{z\=c}^{z\=C}{\int }_{y\=y\left(z\right)}^{y\=Y\left(z\right)}{\int }_{x\=x\left(y\,z\right)}^{x\=X\left(y\,z\right)}f\left(x\,y\,z\right) \mathrm{dx} \mathrm{dy} \mathrm{dz}$ 

${\int }_{y\=b}^{y\=B}{\int }_{z\=z\left(y\right)}^{z\=Z\left(y\right)}{\int }_{x\=x\left(y\,z\right)}^{x\=X\left(y\,z\right)}f\left(x\,y\,z\right) \mathrm{dx} \mathrm{dz} \mathrm{dy}$ 

Table 7.3.1   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

For iteration in the order $\mathrm{dz} \mathrm{dy} \mathrm{dx}$:

The task template at Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Multiple Integration≻Cartesian 3-D

This command is also available through the Context Panel once the Student MultivariateCalculus package has been loaded.

The four relevant task templates  at

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

Over a Cube, Over a Sphere, Over a Tetrahedron, Over a General 3-D Region

The task template at Tools≻Tasks≻Browse:

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

In the Student VectorCalculus package, the modified int command with predefined regions:

Parallelepiped, Sphere, Tetrahedron, Region

The Int and int commands at top-level

Table 7.3.2   Maple tools for iterating a triple integral in Cartesian coordinates

In Examples 7.3.(1 - 6), implement the indicated iteration of the triple integral $\int \int {\int }_{R}1 \mathrm{dv}$ , where $R$ is that portion of the first-octant lying under the cylinder $x^2\+z\=4$ for $y\in \left[0\,3\right]$.

In Examples 7.3.(7 - 12), implement the indicated iteration of the triple integral $\int \int {\int }_{R}1 \mathrm{dv}$ , where $R$ is that portion of the first-octant lying under the cylinder $x^2\+z^2\=1$, but in front of the plane $y\=x$.

Example 7.3.1

$\mathrm{dz} \mathrm{dy} \mathrm{dx}$ 

Example 7.3.7

$\mathrm{dz} \mathrm{dy} \mathrm{dx}$ 

Example 7.3.2

$\mathrm{dz} \mathrm{dx} \mathrm{dy}$ 

Example 7.3.8

$\mathrm{dz} \mathrm{dx} \mathrm{dy}$ 

Example 7.3.3

$\mathrm{dy} \mathrm{dx} \mathrm{dz}$ 

Example 7.3.9

$\mathrm{dy} \mathrm{dx} \mathrm{dz}$ 

Example 7.3.4

$\mathrm{dy} \mathrm{dz} \mathrm{dx}$ 

Example 7.3.10

$\mathrm{dy} \mathrm{dz} \mathrm{dx}$ 

Example 7.3.5

$\mathrm{dx} \mathrm{dy} \mathrm{dz}$ 

Example 7.3.11

$\mathrm{dx} \mathrm{dy} \mathrm{dz}$ 

Example 7.3.6

$\mathrm{dx} \mathrm{dz} \mathrm{dy}$ 

Example 7.3.12

$\mathrm{dx} \mathrm{dz} \mathrm{dy}$ 

Example 7.3.13

$R$ is that portion of the first-octant lying under $z\=x^2\+y^2$, between the planes $y\=x$ and $x\=2$.

Example 7.3.14

$R$ is that portion of the first-octant lying under the plane $x\+y\+2 z\=2$.

Example 7.3.15

$R$ is the region bounded above by $z\=4-x^2-y^2$ and below by $z\=0$.

Example 7.3.16

$R$ is the region bounded above by $z\=x^2\+y^2$ and below by $z\=0$, and lying inside the cylinder $x^2\+y^2\=4$.

Example 7.3.17

$R$ is the region bounded above by $z\=y^2$ and below by $z\=0$, and bounded by the planes $x\=0$, $x\=1$, $y\=-1$, $y\=1$.

Example 7.3.18

$R$ is that portion of the first-octant bounded by the coordinate planes, $x^2\+y^2\=4$, and $x\+z\=3$.

Example 7.3.19

$R$ is the region bounded above by $z\=6-x^2\/2-2 y^2$ and below by $z\=x^2\+4 y^2$.

Example 7.3.20

$R$ is that portion of the first-octant bounded above by $x^2\+z\=1$, and on the right by $y\=x^2\+z^2$.