TRIPLE INTEGRALS | Improve Society

The above results are easily generalized to closed regions in three dimensions. For example, consider a function $F(x, y, z)$ defined in a closed three dimensional region $\cal R$ . Subdivided the region into n subregions of volume $\Delta V_k,\ k = 1,\ 2,\ \dots,\ n$ . Letting $(\xi_k, \eta_k, \zeta_k)$ be some point in each subregion, we form
$\displaystyle \lim\limits_{n\rightarrow\infty}\sum_{k=1}^{n}F(\xi_k, \eta_k, \zeta_k)\Delta V_k\cdots(6)$
where the number n of subdivisions approaches infinity in such a way that the largest linear dimension of each subregion approaches zero. If this limit exists we denote it by
$\displaystyle \underset{\cal R}{\iiint}F(x, y, z)dV\cdots(7)$
called the triple integral of $F(x, y, z)$ over $\cal R$ . The limit dose exist if $F(x, y, z)$ is continuous (or piecewise continuous) in $\cal R$ .

If we construct a grid consisting of planes parallel to the xy, yz and xz planes, the region $\cal R$ is subdivided into subregions which are rectangular parallelepipeds. In such case we can express the triple integral over $\cal R$ given by (7) as an iterated integral of the form
$\displaystyle \int_{x=a}^{b}\int_{y=g_1(x)}^{g_2(x)}\int_{z=f_1(x,y)}^{f_2(x,y)}F(x, y, z)dxdydz = \\ \int_{x=a}^{b} \left [ \int_{y=g_1(x)}^{g_2(x)} \left \{ \int_{z=f_1(x,y)}^{f_2(x,y)} F(x, y, z)dz \right \} dy \right ] dx\cdots(8)$
(where the innermost integral is to be evaluated first) or the sum of such integrals. The integration can also be performed in any other order to give an equivalent result.

Extensions to higher dimensions are also possible.