Triple Integral — Definition, Formula & Examples

A triple integral is the integral of a function of three variables over a three-dimensional region. It generalizes the double integral by adding a third layer of integration, and is commonly used to compute volumes, masses, and other accumulated quantities in 3D.

Given a continuous function $f(x, y, z)$ defined on a bounded closed region $E \subseteq \mathbb{R}^3$ , the triple integral $\iiint_E f(x,y,z)\, dV$ is defined as the limit of Riemann sums $\lim_{\|P\|\to 0} \sum_{i} f(x_i^*, y_i^*, z_i^*) \,\Delta V_i$ , where $P$ is a partition of $E$ into subregions of volume $\Delta V_i$ and $(x_i^*, y_i^*, z_i^*)$ is a sample point in each subregion.

Key Formula

\iiint_E f(x,y,z)\, dV = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z)\, dz\, dy\, dx

Where:

$E$ = The three-dimensional region of integration
$f(x,y,z)$ = The integrand, a function of three variables
$dV$ = The volume element (e.g., dx dy dz in Cartesian coordinates)
$a, b$ = Bounds on the outermost variable x
$g_1, g_2$ = Bounds on y, which may depend on x
$h_1, h_2$ = Bounds on z, which may depend on x and y

How It Works

You evaluate a triple integral by performing three successive single integrals, integrating from the innermost variable outward. The limits of each integral may depend on the outer variables, reflecting the shape of the region. In Cartesian coordinates,

dV = dx\,dy\,dz

. For regions with spherical or cylindrical symmetry, switching to spherical coordinates (

dV = \rho^2 \sin\phi\, d\rho\, d\phi\, d\theta

) or cylindrical coordinates (

dV = r\, dr\, d\theta\, dz

) often simplifies the computation. When

f(x,y,z) = 1

, the triple integral gives the volume of the region

E

Worked Example

Problem: Find the volume of the solid box defined by

0 \le x \le 3

0 \le y \le 2

0 \le z \le 4

Set up the integral: Since we want volume, set

f = 1

and write the triple integral with the given constant bounds.

V = \int_0^3 \int_0^2 \int_0^4 dz\, dy\, dx

Integrate with respect to z: The innermost integral evaluates to the length of the z-interval.

\int_0^4 dz = 4

Integrate with respect to y and x: Multiply through the remaining integrals, each contributing its interval length.

V = \int_0^3 \int_0^2 4\, dy\, dx = \int_0^3 8\, dx = 24

Answer: The volume of the box is

24

cubic units.

Why It Matters

Triple integrals are essential in multivariable calculus, physics, and engineering for computing mass with variable density, moments of inertia, and fluid flow through volumes. The Divergence Theorem converts certain surface integrals into triple integrals, making them a key tool in electromagnetism and fluid dynamics.

Common Mistakes

Mistake: Forgetting the Jacobian factor when changing coordinates (e.g., using

dr\,d\theta\,dz

instead of

r\,dr\,d\theta\,dz

in cylindrical coordinates).

Correction: Always include the correct volume element:

r

for cylindrical and

\rho^2 \sin\phi

for spherical coordinates. This factor accounts for how volume scales in curvilinear coordinates.

Related Terms

Jacobian — Scale factor when changing integration coordinates
Divergence Theorem — Converts surface integrals to triple integrals
Divergence — Integrand in the Divergence Theorem
Partial Derivative — Differentiation used to set up bounds
Green's Theorem — 2D analog linking line and double integrals
Laplacian — Second-order operator often integrated over volumes