Triple integrals are defined for functions of three variables as the limit of a sum, just as the single and double integrals are defined respectively for functions of one and two variables. The Box interval B
B = { x, y, z | a = x = b, c = y = d, and p = z = q} where the function f(x, y, z) is defined is partitioned into lmn sub boxes by dividing the intervals [a, b], [c, d] and [p. q] into l, m and n sub intervals of equal widths $\Delta$x, $\Delta$y and $\Delta$z for respective partitions.

Triple Integral

Each sub box has thus a volume $\Delta$V = $\Delta$x. $\Delta$y. $\Delta$z. Analogically with single and double integrals, the triple integral of f(x, y, z) over the Box is defined as the limit of the triple Riemann sum, if the limit exists.$\int \int_{B} \int f(x,y,z)dV$ = $\lim_{l,m,n \to \infty }\sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}f(x_{ijk}y_{ijk}z_{ijk})\Delta V$.

For practical purpose, Fubini's theorem on triple integral is used to evaluate triple integrals.
If f(x, y, z) is continuous on a rectangular Box B = [a, b] x [c, d] x [p, q], then
$\int \int_{B} \int f(x,y,z)dV$ = $\int_{p}^{q}\int_{c}^{d}\int_{a}^{b}f(x,y,z)dxdydz$
If the solid region considered for triple integrals happens to lie between the graphs two continuous functions of any two of the variables, the formula used is modified to include a double integral as follows:
$\int \int_{E} \int f(x,y,z)dV$ = $\int \int_{D}$$[\int_{u_{1}(x,y)}^{u_{2}(x,y)}f(x,y,z)dz]dA$
where the region E is bounded by the continuous functions z = u1 (x,y) and z =u2 (x, y).
Similar formulas are also used when the solid region for the triple integrals is bounded by functions of other two variable combinations like yz and zx.

The triple integral cannot be interpreted geometrically as the function f in three variables represents a hyper surface in four dimensional space and cannot be visualized.

Suppose the solid region on which the triple integral is defined is
E = {(x, y, z) | (x, y) ∈ D, u1 (x,y) ≤ z ≤ u2 (x,y)} where the plane region D is defined in polar coordinates as
D = { (r, θ) | α ≤ θ ≤ β, h1 (θ) ≤ r ≤ h2 (θ) }.
The triple integral in rectangular coordinates
$\underset{E}{\int \int \int }f(x,y,z)dV$ = $\underset{D}{\int \int }[\int_{u_{1}(x,y)}^{u_{2}(x,y)}f(x,y,z)dz]dA$
can be transformed for cylindrical coordinates system as
$\underset{E}{\int \int \int }f(x,y,z)dV$ = $\int_{\alpha }^{\beta }\int_{h_{1}(\theta )}^{h_{2}(\theta )}\int_{u_{1}(rcos\theta ,rsin\theta )}^{u_{2}(rcos\theta ,rsin\theta )}f(rcos\theta ,rsin\theta ,z)rdzdrd\theta $.

Remember the transformations required for cylindrical coordinate system are x = r cos θ, y = r sin θ, z = z and
dV = r dz dr dθ.
This transformation is useful when the solid region E can be easily described in cylindrical coordinates and the f (x, y, z) contains expressions of the form x2 + y2.