Definition and Properties of Triple Integrals

Definition of Triple Integral

We can introduce the triple integral similar to double integral as a limit of a Riemann sum. We start from the simplest case when the region of integration U is a rectangular box [a, b] × [c, d] × [p, q] (Figure 1).

Let the set of numbers $$\left\{ {{x_0},{x_1}, \ldots ,{x_m}} \right\}$$ be a partition of $$\left[ {a,b} \right]$$ into small intervals so that the following relations are valid:

$a = {x_0} \lt {x_0} \lt {x_1} \lt {x_2} \lt \ldots \lt {x_i} \lt \ldots \lt {x_{m - 1}} \lt {x_m} = b.$

Similarly, we can construct partitions of the segment $$\left[ {c,d} \right]$$ along the $$y$$-axis and the segment $$\left[ {p,q} \right]$$ along the $$z$$-axis:

$c = {y_0} \lt {y_1} \lt {y_2} \lt \ldots \lt {y_j} \lt \ldots \lt {y_{n - 1}} \lt {y_n} = d,$
$p = {z_0} \lt {z_1} \lt {z_2} \lt \ldots \lt {z_k} \lt \ldots \lt {z_{\ell - 1}} \lt {z_\ell} = q.$

The Riemann sum of the function $$f\left( {x,y,z} \right)$$ over the partition of $$\left[ {a,b} \right] \times \left[ {c,d} \right]$$ $$\times \left[ {p,q} \right]$$ is defined by

$\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {\sum\limits_{k = 1}^{\ell} {f\left( {{u_i},{v_j},{w_k}} \right) \Delta {x_i}\Delta {y_j}\Delta {z_k}} } } .$

Here $${\left( {{u_i},{v_j},{w_k}} \right)}$$ is some point in the rectangular box $$\left( {{x_{i - 1}},{x_i}} \right)$$ $$\times \left( {{y_{j - 1}},{y_j}} \right)$$ $$\times \left( {{z_{k - 1}},{z_k}} \right),$$ and the differences are

$\Delta {x_i} = {x_i} - {x_{i - 1}},\;\; \Delta {y_j} = {y_j} - {y_{j - 1}},\;\; \Delta {z_k} = {z_k} - {z_{k - 1}}.$

The triple integral of a function $$f\left( {x,y,z} \right)$$ in the parallelepiped $$\left[ {a,b} \right] \times \left[ {c,d} \right] \times \left[ {p,q} \right]$$ is defined as a limit of the Riemann sum, such that the maximum values of the differences $$\Delta {x_i},$$ $$\Delta {y_j}$$ and $$\Delta {z_k}$$ approach zero:

$\iiint\limits_{\left[ {a,b} \right] \times \left[ {c,d} \right] \times \left[ {p,q} \right]} {f\left( {x,y,z} \right)dV = } \lim\limits_{\substack{ \text{max}\,\Delta {x_i} \to 0\\ \text{max}\,\Delta {y_j} \to 0\\ \text{max}\,\Delta {z_k} \to 0}} {\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {\sum\limits_{k = 1}^\ell} {f\left( {{u_i},{v_j},{w_k}} \right) }} \Delta {x_i}\Delta {y_j}\Delta {z_k}} .$

To define the triple integral over a general region $$U,$$ we choose a rectangular box $$\left[ {a,b} \right]$$ $$\times \left[ {c,d} \right]$$ $$\times \left[ {p,q} \right]$$ containing the given region $$U.$$ Then we introduce the function $$g\left( {x,y,z} \right)$$ such that

$\begin{cases} {g\left( {x,y,z} \right) = f\left( {x,y,z} \right),} \text{ if}\;f \in U \\ {g\left( {x,y,z} \right) = 0,} \text{ if}\;f \notin U \end{cases}.$

Then the triple integral of the function $$f\left( {x,y,z} \right)$$ over a general region $$U$$ is defined as

$\iiint\limits_U {f\left( {x,y,z} \right)dV} = \iiint\limits_{\left[ {a,b} \right] \times \left[ {c,d} \right] \times \left[ {p,q} \right]} {g\left( {x,y,z} \right)dV} .$

Properties of Triple Integrals

Let $$f\left( {x,y,z} \right)$$ and $$g\left( {x,y,z} \right)$$ be functions which are integrable in the region $$U.$$ Then the following properties are valid:

1. $${\iiint\limits_U {\left[ {f\left( {x,y,z} \right) + g\left( {x,y,z} \right)} \right]dV} }$$ $$= {\iiint\limits_U {f\left( {x,y,z} \right)dV} }$$ $$+{ \iiint\limits_U {g\left( {x,y,z} \right)dV} ;}$$
2. $${\iiint\limits_U {\left[ {f\left( {x,y,z} \right) - g\left( {x,y,z} \right)} \right]dV} }$$ $$= {\iiint\limits_U {f\left( {x,y,z} \right)dV} }$$ $$-{ \iiint\limits_U {g\left( {x,y,z} \right)dV} ;}$$
3. $${\iiint\limits_U {kf\left( {x,y,z} \right)dV} }$$ $$={ k\iiint\limits_U {f\left( {x,y,z} \right)dV},}$$ where $$k$$ is a constant;
4. If $${f\left( {x,y,z} \right)} \le {g\left( {x,y,z} \right)}$$ at any point of the region $$U,$$ then
$\iiint\limits_U {f\left( {x,y,z} \right)dV} \le \iiint\limits_U {g\left( {x,y,z} \right)dV} ;$
5. If the region $$U$$ is a union of two non-overlapping regions $${U_1}$$ and $${U_2},$$ then
$\iiint\limits_U {f\left( {x,y,z} \right)dV} = \iiint\limits_{{U_1}} {f\left( {x,y,z} \right)dV} + \iiint\limits_{{U_2}} {f\left( {x,y,z} \right)dV} ;$
6. Let $$m$$ be the minimum and $$M$$ be the maximum value of a continuous function $$f\left( {x,y,z} \right)$$ in the region $$U.$$ Then the following estimate is valid for the triple integral:
$m \cdot V \le \iiint\limits_U {f\left( {x,y,z} \right)dV} \le M \cdot V,$
where $$V$$ is the volume of the integration region $$U.$$
7. The Mean Value Theorem for Triple Integrals
If a function $$f\left( {x,y,z} \right)$$ is continuous in the region $$U,$$ then there exists a point $${M_0} \in U$$ such that
$\iiint\limits_U {f\left( {x,y,z} \right)dV} = f\left( {{M_0}} \right) \cdot V,$
where $$V$$ is the volume of the region $$U.$$

See solved problems on Page 2.