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

$a\=a\left(x\,y\,z\right)$ 

$b\=b\left(x\,y\,z\right)$ 

$c\=c\left(x\,y\,z\right)$ 

$x\=x\left(a\,b\,c\right)$ 

$y\=y\left(a\,b\,c\right)$ 

$z\=z\left(a\,b\,c\right)$ 

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

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

${\int }_{\mathrm{\θ}\=a}^{\mathrm{\θ}\=A}{\int }_{\mathrm{\φ}\=\mathrm{\φ}\left(\mathrm{\θ}\right)}^{\mathrm{\φ}\=\mathrm{\Φ}\left(\mathrm{\θ}\right)}{\int }_{\mathrm{\ρ}\=\mathrm{\ρ}\left(\mathrm{\φ}\,\mathrm{\θ}\right)}^{\mathrm{\ρ}\=\mathrm{\Ρ}\left(\mathrm{\φ}\,\mathrm{\θ}\right)}f\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right) {\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) d\mathrm{\ρ} d\mathrm{\φ} d\mathrm{\θ}$ 

${\int }_{\mathrm{\φ}\=b}^{\mathrm{\φ}\=B}{\int }_{\mathrm{\θ}\=\mathrm{\θ}\left(\mathrm{\φ}\right)}^{\mathrm{\θ}\=\mathrm{\Θ}\left(\mathrm{\φ}\right)}{\int }_{\mathrm{\ρ}\=\mathrm{\ρ}\left(\mathrm{\φ}\,\mathrm{\θ}\right)}^{\mathrm{\ρ}\=\mathrm{\Ρ}\left(\mathrm{\φ}\,\mathrm{\θ}\right)}f\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right) {\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) d\mathrm{\ρ} d\mathrm{\θ} d\mathrm{\φ}$ 

${\int }_{\mathrm{\θ}\=a}^{\mathrm{\θ}\=A}{\int }_{\mathrm{\ρ}\=\mathrm{\ρ}\left(\mathrm{\θ}\right)}^{\mathrm{\ρ}\=\mathrm{\Ρ}\left(\mathrm{\θ}\right)}{\int }_{\mathrm{\φ}\=\mathrm{\φ}\left(\mathrm{\ρ}\,\mathrm{\θ}\right)}^{\mathrm{\φ}\=\mathrm{\Φ}\left(\mathrm{\ρ}\,\mathrm{\θ}\right)}f\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right) {\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) d\mathrm{\φ} d\mathrm{\ρ} d\mathrm{\θ}$ 

${\int }_{\mathrm{\ρ}\=c}^{\mathrm{\ρ}\=C}{\int }_{\mathrm{\θ}\=\mathrm{\θ}\left(\mathrm{\ρ}\right)}^{\mathrm{\θ}\=\mathrm{\Θ}\left(\mathrm{\ρ}\right)}{\int }_{\mathrm{\φ}\=\mathrm{\φ}\left(\mathrm{\ρ}\,\mathrm{\θ}\right)}^{\mathrm{\φ}\=\mathrm{\Φ}\left(\mathrm{\ρ}\,\mathrm{\θ}\right)}f\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right) {\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) d\mathrm{\φ} d\mathrm{\θ} d\mathrm{\ρ}$ 

${\int }_{\mathrm{\φ}\=b}^{\mathrm{\φ}\=B}{\int }_{\mathrm{\ρ}\=\mathrm{\ρ}\left(\mathrm{\φ}\right)}^{\mathrm{\ρ}\=\mathrm{\Ρ}\left(\mathrm{\φ}\right)}{\int }_{\mathrm{\θ}\=\mathrm{\θ}\left(\mathrm{\ρ}\,\mathrm{\φ}\right)}^{\mathrm{\θ}\=\mathrm{\Θ}\left(\mathrm{\ρ}\,\mathrm{\φ}\right)}f\left(r\,\mathrm{\φ}\,\mathrm{\θ}\right) {\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) d\mathrm{\θ} d\mathrm{\ρ} d\mathrm{\φ}$ 

${\int }_{\mathrm{\ρ}\=c}^{\mathrm{\ρ}\=C}{\int }_{\mathrm{\φ}\=\mathrm{\φ}\left(\mathrm{\ρ}\right)}^{\mathrm{\φ}\=\mathrm{\Φ}\left(\mathrm{\ρ}\right)}{\int }_{\mathrm{\θ}\=\mathrm{\θ}\left(\mathrm{\ρ}\,\mathrm{\φ}\right)}^{\mathrm{\θ}\=\mathrm{\Θ}\left(\mathrm{\ρ}\,\mathrm{\φ}\right)}f\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right) {\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) d\mathrm{\θ} d\mathrm{\φ} d\mathrm{\ρ}$ 

Table 7.6.1   In spherical 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 ${\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)$ must be included in the integrand.

In the Student MultivariateCalculus package, the MultiInt command with the option "coordinates = spherical" if the coordinate names are $r\,\mathrm{\φ}\,\mathrm{\θ}$ or "coordinates = spherical[$n_1\,n_2\,n_3$]" if the coordinate names are $n_1\,n_2\,n_3$. (Note the defaulting to $r$ rather than to $\mathrm{\ρ}$.)

The task template at Tools≻Tasks≻Browse:

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

For iteration in the order $d\mathrm{\ρ} d\mathrm{\φ} d\mathrm{\θ}$, with exactly those variable names:

The task template at Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Multiple Integration≻Spherical

The Int and int commands at top-level

The Jacobian ${\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)$ must be included in the integral

Table 7.6.2   Maple tools for iterating a triple integral in spherical coordinates

Example 7.6.1

$R$ is the region above the plane $z\=3\/4$ in a unit sphere with center at the origin.

(This region was graphed in Example 7.5.3.)

Example 7.6.2

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

(This region was graphed in Example 7.5.4.)

(See Example 7.4.11 where this integral is evaluated in cylindrical coordinates.)

Example 7.6.3

$R$ is the region inside the unit sphere centered at the origin, and outside the cone $z\=\sqrt{x^2\+y^2}$.

(This region was graphed in Example 7.5.5.)

Example 7.6.4

$R$ is the region bounded below by the cone $\mathrm{\φ}\=\mathrm{\π}\/6$ and above by the sphere $\mathrm{\ρ}\=2$.

(This region was graphed in Example 7.5.6.)

Example 7.6.5

$R$ is the region between the spheres $\mathrm{\ρ}\=1$ and $\mathrm{\ρ}\=2$, and above the cone $\mathrm{\φ}\=\mathrm{\π}\/3$.

(This region was graphed in Example 7.5.7.)

Example 7.6.6

$R$ is the region above the cone $\mathrm{\φ}\=\mathrm{\π}\/4$ and below the sphere $\mathrm{\ρ}\=3 \cos \left(\mathrm{\φ}\right)$.

(This region was graphed in Example 7.5.8.)

Example 7.6.7

$R$ is the region enclosed by the surface $\mathrm{\ρ}\=1-\cos \left(\mathrm{\φ}\right)$.

(This region was graphed in Example 7.5.9.)

Example 7.6.8

$R$ is the region bounded inside by the surface $\mathrm{\ρ}\=1\+\cos \left(\mathrm{\φ}\right)$ and outside by the sphere $\mathrm{\ρ}\=2$.

(This region was graphed in Example 7.5.10.)

Example 7.6.9

If $x\=\mathrm{\ρ} \cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\,y\=\mathrm{\ρ} \sin \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\,z\=\mathrm{\ρ} \cos \left(\mathrm{\φ}\right)$, show that $\frac{\partial \left(x\,y\,z\right)}{\partial \left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)}\={\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)$.