Working in spherical coordinates $\mathrm{\ρ}\,\mathrm{\φ}$, and $\mathrm{\θ}$, where $x\=\mathrm{\ρ} \sin \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)$, $y\=\mathrm{\ρ} \sin \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)$, $z\=\mathrm{\ρ} \cos \left(\mathrm{\φ}\right)$, select a volume element $\mathrm{dv}$. There are six possible choices, namely, ${\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)$ times any one of the following:

$d\mathrm{\ρ} d\mathrm{\φ} d\mathrm{\θ}$, $d\mathrm{\ρ} d\mathrm{\θ} d\mathrm{\φ}$, $d\mathrm{\φ} d\mathrm{\ρ} d\mathrm{\θ}$, $d\mathrm{\φ} d\mathrm{\θ} d\mathrm{\ρ}$, $d\mathrm{\θ} d\mathrm{\φ} d\mathrm{\ρ}$, $d\mathrm{\θ} d\mathrm{\ρ} d\mathrm{\φ}$ 

Enter an integrand and the appropriate bounds of integration.

Compute the value of the integral either exactly or numerically.

Obtain a graph of the 3-D region determined by the limits of integration. The bounding faces for the region of integration are drawn with the following color-coding: ${\int }_{\mathrm{yellow}}^{\mathrm{gray}}{\int }_{\mathrm{green}}^{\mathrm{brown}}{\int }_{\mathrm{blue}}^{\mathrm{red}}\mathrm{\Psi } \mathit{\ⅆ}v$.

Int, plot3d, plots[display]

Multivariate Calculus > Integration > Multiple Integration > Spherical

Student[MultivariateCalculus], VectorCalculus[int]