Triple Integration

The triple integral of $f(x,y,z)$ over the region $R$ is defined as:

$$\int \int \int f(x,y,z)dV=\underset{||\Delta V_i||\to 0}{\lim }\sum _{i=1}^{n}f(x_i^{\ast },y_i^{\ast },z_i^{\ast })\Delta V_i$$

where $\{R_i|i\in I\}$ is a partition of $R$, $\Delta V_i$ is the volume of the sub-region $R_i$ and $(x_i^{\ast },y_i^{\ast },z_i^{\ast })\in R_i$.

Like Double Integrals, we say the triple integral exists exactly when the limit exists.

Theorem

Let $S$ be a closed, piecewise-smooth surface that encloses a region $R$ with finite volume. If $f(x,y,z)$ is a continuous function inside and on $V$, then the triple integral of $f(x,y,z)$ over $R$ exists.

Triple Iterated Integrals

A triple iterated integral is an integral of the form:

$${\int }_a^b\left({\int }_{g_1(x)}^{g_2(x)}\left({\int }_{h_1(x,y)}^{h_2(x,y)}f(x,y)dz\right)dy\right)dx$$

The ordering of the variables of integration can also change as long as the limit functions change appropriately.