Summary of Triple Integrals

To compute a triple integral we use Fubini’s theorem, which states that if [latex]f(x,y,z)[/latex] is continuous on a rectangular box
To compute the volume of a general solid bounded region [latex]E[/latex] 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
[latex]\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[/latex]

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