Definition and Properties of Triple Integrals

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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 \(\left[ {a,b} \right] \times \left[ {c,d} \right] \) \(\times \left[ {p,q} \right]\) (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:

\({\iiint\limits_U {\left[ {f\left( {x,y,z} \right) + g\left( {x,y,z} \right)} \right]dV} }\) \(= {\iiint\limits_U {f\left( {x,y,z} \right)dV} }\) \(+{ \iiint\limits_U {g\left( {x,y,z} \right)dV} ;}\)
\({\iiint\limits_U {\left[ {f\left( {x,y,z} \right) - g\left( {x,y,z} \right)} \right]dV} }\) \(= {\iiint\limits_U {f\left( {x,y,z} \right)dV} }\) \(-{ \iiint\limits_U {g\left( {x,y,z} \right)dV} ;}\)
\({\iiint\limits_U {kf\left( {x,y,z} \right)dV} }\) \(={ k\iiint\limits_U {f\left( {x,y,z} \right)dV},}\) where \(k\) is a constant;
If \({f\left( {x,y,z} \right)} \le {g\left( {x,y,z} \right)}\) at any point of the region \(U,\) then
\[\iiint\limits_U {f\left( {x,y,z} \right)dV} \le \iiint\limits_U {g\left( {x,y,z} \right)dV} ;\]
If the region \(U\) is a union of two non-overlapping regions \({U_1}\) and \({U_2},\) then
\[\iiint\limits_U {f\left( {x,y,z} \right)dV} = \iiint\limits_{{U_1}} {f\left( {x,y,z} \right)dV} + \iiint\limits_{{U_2}} {f\left( {x,y,z} \right)dV} ;\]
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 \(U.\)
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 \(U.\)