Key Equations

• Triple integral in cylindrical coordinates
  $\underset{B}{\displaystyle\iiint} g(x,y,z)dV=\underset{B}{\displaystyle\iiint} g(r\cos\theta,r\sin\theta,z)r dr d\theta dz=\underset{B}{\displaystyle\iiint} f(r,\theta,z)r dr d\theta dz=$
• Triple integral in spherical coordinates
  $\underset{B}{\displaystyle\iiint} f(\rho,\theta,\varphi)\rho^{2}\sin\varphi{d}\rho{d}\varphi{d}\theta=\displaystyle\int_{\varphi=\gamma}^{\varphi=\psi}\displaystyle\int_{\theta=\alpha}^{\theta=\beta}\displaystyle\int_{\rho=a}^{\rho=b} \rho^{2}\sin\varphi{d}\rho{d}\varphi{d}\theta$

Glossary

triple integral in cylindrical coordinates

the limit of a triple Riemann sum, provided the following limit exists:${\displaystyle\lim_{l,m,n\to\infty}{\sum_{i=1}^{l}}{\displaystyle\sum_{j=1}^{m}}{\displaystyle\sum_{k=1}^{n}f({r^{*}}_{i,j,k}, {{\theta}^{*}}_{i,j,k}, {{z}^{*}}_{i,j,k}){r^{*}}_{i,j,k}{\Delta}r{\Delta}{\theta}{\Delta}{z}}}$

triple integral in spherical coordinates

the limit of a triple Riemann sum, provided the following limit exists: ${\displaystyle\lim_{l,m,n\to\infty}{\displaystyle\sum_{i=1}^{l}}{\displaystyle\sum_{j=1}^{m}}{\displaystyle\sum_{k=1}^{n}f({{\rho}^{*}}_{i,j,k}, {{\theta}^{*}}_{i,j,k}, {{\varphi}^{*}}_{i,j,k})({{\rho}^{*}}_{i,j,k})^{2}\sin{\varphi}{\Delta}{\rho}{\Delta}{\theta}{\Delta}{\varphi}}}$