Summary of Triple Integrals

To compute a triple integral we use Fubini’s theorem, which states that if $f(x,y,z)$ is continuous on a rectangular box
To compute the volume of a general solid bounded region $E$ we use the triple integral
Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.
To compute the average value of a function over a general three-dimensional region, we use

Key Equations

Triple integral
$\underset{l,m,n\to\infty}{\lim}\displaystyle\sum_{i=1}^{l}\displaystyle\sum_{j=1}^{m}\displaystyle\sum_{k=1}^{n} f(x_{ijk}^{\ast},y_{ijk}^{\ast},z_{ijk}^{\ast}) \Delta{x}\Delta{y}\Delta{z}=\underset{B}{\displaystyle\iiint} f(x,y,z)dV$

triple integral: the triple integral of a continuous function $f(x, y, z)$ over a rectangular solid box $\bf{B}$ is the limit of a Riemann sum for a function of three variables, if this limit exists