Triple Integral

An integral over 3 variables.

$${\iiint }_{E}f(\vec{x})dV$$

Cartesian

$${\iiint }_{E}f(x,y,z)dxdydz$$

where $x$ is the amount a point lies in the positive $x$ direction where $y$ is the amount a point lies in the positive $y$ direction where $z$ is the amount a point lies in the positive $z$ direction

Cylindrical

Polar Coordinates + $z$

$${\iiint }_{E}f(r,\theta ,z)\cdot rdrd\theta dz$$

where the Jacobian Determinant $\det J_f=r$ where $r$ is the distance of a point’s $xy$ Projection from the origin where $\theta$ is the angle of the point from the positive $x$ axis where $z$ is the amount a point lies in the positive $z$ direction

Spherical

$${\iiint }_{E}f(\rho ,\theta ,\varphi )\cdot {\rho }^2\sin (\varphi )drd\theta dz$$

where the Jacobian Determinant $\det J_f={\rho }^2\sin (\varphi )$ where $\rho$ is the distance of a point from the origin where $\theta$ is the angle of the point from the positive $x$ axis where $\varphi$ is the angle of the point from the positive $z$ axis