List of basic formulas related to triple integrals:
- Definition of a triple integral:
[
\iiint_V f(x, y, z) \, dV
]
where ( V ) is the volume in which the function ( f(x, y, z) ) integrates. - Order of integration:
The triple integral can be computed in any order. For example:
[
\iiint_V f(x, y, z) \, dV = \int_{z_1}^{z_2} \int_{y_1}^{y_2} \int_{x_1}^{x_2} f(x, y, z) \, dx \, dy \,
d - Parametric limits:
If the integration limits depend on other variables, then:
[
\iiint_V f(x, y, z) \, dV = \int_{z_1}^{z_2} \int_{y_1(z)}^{y_2(z )} \int_{x_1(y,z)}^{x_2(y,z)} f(x, y, z) \, dx \, dy \,
dz - Conversion to cylindrical coordinates:
[
\iiint_V f(r, \theta, z) \, r \, dr \, d\theta \, dz
]
where ( r ) is the radius, ( \theta ) is the angle, ( z height. - Conversion to spherical coordinates:
[
\iiint_V f(\rho, \theta, \phi) \, \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi
]
where ( \ rho ) is the radius, ( \theta ) is the angle in the xy plane, ( \phi ) is the angle from the z axis.