Working in cylindrical coordinates $r\,\mathrm{\θ}$, and $z$, select a volume element $\mathrm{dv}$. (There are six possible choices: $r \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$, $r \mathrm{dz} d\mathrm{\θ} \mathrm{dr}$, $r \mathrm{dr} d\mathrm{\θ} \mathrm{dz}$, $r \mathrm{dr} \mathrm{dz} d\mathrm{\θ}$, $r d\mathrm{\θ}\mathit{ }\mathrm{dr} \mathrm{dz}$, $r d\mathrm{\θ} \mathrm{dz} \mathrm{dr}$.)

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 > Cylindrical

Student[MultivariateCalculus], Student[VectorCalculus][int]