$\int \int {\int }_{R}f\left(x\,y\,z\right) \mathrm{dv}$ = $\int \int {\int }_{R\prime }f\left(x\left(a\,b\,c\right)\,y\left(a\,b\,c\right)\,z\left(a\,b\,c\right)\right) \left|\frac{\partial \left(x\,y\,z\right)}{\partial \left(a\,b\,c\right)}\right| \mathrm{dv}\prime$

$a\=a\left(x\,y\,z\right)$ 

$b\=b\left(x\,y\,z\right)$ 

$c\=c\left(x\,y\,z\right)$ 

$x\=x\left(a\,b\,c\right)$ 

$y\=y\left(a\,b\,c\right)$ 

$z\=z\left(a\,b\,c\right)$ 

$\left|\frac{\partial \left(x\,y\,z\right)}{\partial \left(a\,b\,c\right)}\right|$ =  $\|\begin{array}{ccc}{x}_a& x_b& x_c\\ y_a& y_b& y_c\\ z_a& z_b& z_c\end{array}\|$ 

$x\=r \cos \left(\mathrm{\θ}\right)\,y\=r \sin \left(\mathrm{\θ}\right)\,z\=z$ 

${\int }_{\mathrm{\θ}\=a}^{\mathrm{\θ}\=A}{\int }_{r\=r\left(\mathrm{\θ}\right)}^{r\=R\left(\mathrm{\θ}\right)}{\int }_{z\=z\left(r\,\mathrm{\θ}\right)}^{z\=Z\left(r\,\mathrm{\θ}\right)}f\left(r\,\mathrm{\θ}\,z\right) r \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ 

${\int }_{r\=b}^{r\=B}{\int }_{\mathrm{\θ}\=\mathrm{\θ}\left(r\right)}^{\mathrm{\θ}\=\mathrm{\Θ}\left(r\right)}{\int }_{z\=z\left(r\,\mathrm{\θ}\right)}^{z\=Z\left(r\,\mathrm{\θ}\right)}f\left(r\,\mathrm{\θ}\,z\right) r \mathrm{dz} d\mathrm{\θ} \mathrm{dr}$ 

${\int }_{z\=c}^{z\=C}{\int }_{\mathrm{\θ}\=\mathrm{\θ}\left(z\right)}^{\mathrm{\θ}\=\mathrm{\Θ}\left(z\right)}{\int }_{r\=r\left(\mathrm{\θ}\,z\right)}^{r\=R\left(\mathrm{\θ}\,z\right)}f\left(r\,\mathrm{\θ}\,z\right) r \mathrm{dr} d\mathrm{\θ} \mathrm{dz}$ 

${\int }_{\mathrm{\θ}\=a}^{\mathrm{\θ}\=A}{\int }_{z\=z\left(\mathrm{\θ}\right)}^{z\=Z\left(\mathrm{\θ}\right)}{\int }_{r\=r\left(\mathrm{\θ}\,z\right)}^{r\=R\left(\mathrm{\θ}\,z\right)}f\left(r\,\mathrm{\θ}\,z\right) r \mathrm{dr} \mathrm{dz} d\mathrm{\θ}$ 

${\int }_{z\=c}^{z\=C}{\int }_{r\=r\left(z\right)}^{r\=R\left(z\right)}{\int }_{\mathrm{\θ}\=\mathrm{\θ}\left(r\,z\right)}^{\mathrm{\θ}\=\mathrm{\Θ}\left(r\,z\right)}f\left(r\,\mathrm{\θ}\,z\right) r d\mathrm{\θ} \mathrm{dr} \mathrm{dz}$ 

${\int }_{r\=b}^{r\=B}{\int }_{z\=z\left(r\right)}^{z\=Z\left(r\right)}{\int }_{\mathrm{\θ}\=\mathrm{\θ}\left(r\,z\right)}^{\mathrm{\θ}\=\mathrm{\Θ}\left(r\,z\right)}f\left(r\,\mathrm{\θ}\,z\right) r d\mathrm{\θ} \mathrm{dz} \mathrm{dr}$ 

Table 7.4.1   In cylindrical 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 Jacobian $r$ must be included in the integrand.

In the Student MultivariateCalculus package, the MultiInt command with the option "coordinates = cylindrical" if the coordinate names are $r\,\mathrm{\θ}\,z$ or "coordinates = cylindrical[$n_1\,n_2\,n_3$]" if the coordinate names are $n_1\,n_2\,n_3$.

The task template at Tools≻Tasks≻Browse:

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

For iteration in the order $\mathrm{dz} \mathrm{dr} d\mathrm{\θ}$:

The task template at Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Multiple Integration≻Cylindrical

The Int and int commands at top-level

The Jacobian $r$ must be included in the integral

Table 7.4.2   Maple tools for iterating a triple integral in cylindrical coordinates

Explicit

$\mathrm{plot3d}\left(f\left(n_1\,n_2\right)\,n_1\=\mathrm{range} 1\, n_2\=\mathrm{range} 2\, \mathrm{coords}\=\mathrm{cylindrical}\right)$

Parametric

$\mathrm{plot3d}\left(\left[r\left(u\,v\right)\,\mathrm{\θ}\left(u\,v\right)\,z\left(u\,v\right)\right]\,u\=\mathrm{range} u\, v\=\mathrm{range} v\,\mathrm{coords}\=\mathrm{cylindrical}\right)$

Table 7.4.3   Syntax by which the plot3d command will graph in cylindrical coordinates

Example 7.4.1

$R$ is the region bounded above by the upper hemisphere in a sphere of radius $1$, and below by the plane through the "equator".

Example 7.4.2

$R$ is the interior of the cylinder $x^2\+y^2\=1\/4$, bounded above by the plane $x\+y\+z\=1$, and below by the plane $z\=0$.

Example 7.4.3

$R$ is the region bounded below by $z\=x^2\+y^2$, above by $z\=1\+x^2\+y^2$, and laterally by the cylinder $x^2\+y^2\=1$.

Example 7.4.4

$R$ is the region bounded above by $z\=5-x^2-y^2$ and below by $z\=4 x^2\+4 y^2$.

Example 7.4.5

$R$ is the region bounded above by the sphere $x^2\+y^2\+z^2\=2$, and below by $z\=x^2\+y^2$.

Example 7.4.6

$R$ is the region outside the cylinder $x^2\+y^2\=1$, bounded above by $z\=9-x^2-y^2$, and below by $z\=0$.

Example 7.4.7

$R$ is the region above $z\=1\+3 x^2\+3 y^2$, below $2 x\+3 y\+ z\=11$, and inside the cylinder $x^2\+y^2\=1$.

Example 7.4.8

$R$ is the region bounded below by the paraboloid $z\=3 x^2\+3 y^2$ and above by the plane $z\=5$.

Example 7.4.9

$R$ is the region inside the cylinder whose cross section is the cardioid $r\=3\+2 \cos \left(\mathrm{\θ}\right)$, bounded above by the plane $x\+2 y\+3 z\=10$, and below by the plane $z\=0$.

Example 7.4.10

$R$ is that part of the interior of the sphere $x^2\+y^2\+z^2\=4$ that lies inside the cylinder whose cross section is $r\=2 \sin \left(\mathrm{\θ}\right)$.

Example 7.4.11

$R$ is the region above the cone $z\=\sqrt{x^2\+y^2}$ but below the unit sphere that is centered at $\left(0\,0\,1\right)$.

Example 7.4.12

$R$ is the region above the plane $z\=0$, below the cone $z\=2\sqrt{x^2\+y^2}$, and inside the cylinder whose cross section is the cardioid $r\=3\+2 \sin \left(\mathrm{\θ}\right)$.

Example 7.4.13

$R$ is the region above the plane $z\=0$, below the plane $z\=x$, inside the cylinder whose cross section is $x^2\+y^2\=1$, and lying in the half-space $x\ge 0$.

Example 7.4.14

$R$ is the region between the planes $z\=0$ and $2 x\+3 y\+5 z\=17$, and above the annulus whose radii are 1 and 2, and whose center is at the origin.

Example 7.4.15

If $x\=r \cos \left(\mathrm{\θ}\right)\,y\=r \sin \left(\mathrm{\θ}\right)\, z\=z$, show that $\frac{\partial \left(x\,y\,z\right)}{\partial \left(r\,\mathrm{\θ}\,z\right)}\=r$.