13.1 Definition and properties of triple integrals

Definition and Notation

Triple Integrals and Notation

Triple integrals extend double integrals into three dimensions, letting you integrate a function over a solid region in space. The notation is:

$\iiint_D f(x, y, z) \, dV$

Here, $f(x, y, z)$ is the integrand (the function you're integrating), $D$ is the three-dimensional solid region you're integrating over, and $dV$ is the volume element, which tells you you're summing up infinitesimal chunks of volume.

Think of it this way: a single integral adds up values along a line, a double integral adds up values over a flat region, and a triple integral adds up values throughout a solid. When $f(x,y,z) = 1$, the triple integral simply gives you the volume of $D$. When $f$ represents a density, the triple integral gives you total mass.

Domain of Integration

The domain $D$ is the solid region over which you integrate. You describe it using inequalities on $x$, $y$, and $z$.

• A rectangular box is the simplest case: $a \leq x \leq b$, $c \leq y \leq d$, $e \leq z \leq f$
• More complex domains include spheres, cylinders, tetrahedra, or solids bounded by arbitrary surfaces
• For non-rectangular domains, the bounds on one or more variables will depend on the other variables (e.g., $0 \leq z \leq 1 - x - y$)

Getting the domain description right is often the hardest part of setting up a triple integral. Sketching the region or at least identifying its bounding surfaces will save you a lot of trouble.

Cartesian Coordinates

In this section, triple integrals are set up in Cartesian coordinates $(x, y, z)$, where the three mutually perpendicular axes meet at the origin $(0, 0, 0)$. The volume element in Cartesian coordinates is:

$dV = dx \, dy \, dz$

Cartesian coordinates work well for domains with flat boundaries (planes, rectangular boxes, tetrahedra). For domains with spherical or cylindrical symmetry, other coordinate systems (covered later) are often more convenient.

Iterated Integrals and Fubini's Theorem

Iterated Integrals

You evaluate a triple integral by converting it into three nested single integrals, called an iterated integral. For a rectangular box domain, this looks like:

$\iiint_D f(x, y, z) \, dV = \int_a^b \int_c^d \int_e^f f(x, y, z) \, dz \, dy \, dx$

To evaluate this:

1. Innermost integral first. Integrate with respect to $z$ (treating $x$ and $y$ as constants), using the $z$-bounds.
2. Middle integral next. Take the result from step 1 and integrate with respect to $y$ (treating $x$ as a constant), using the $y$-bounds.
3. Outermost integral last. Take the result from step 2 and integrate with respect to $x$, using the $x$-bounds.

For non-rectangular domains, the bounds of the inner integrals will be functions of the outer variables rather than constants.

Order of Integration

There are six possible orders of integration for three variables:

$dz\,dy\,dx, \quad dz\,dx\,dy, \quad dy\,dz\,dx, \quad dy\,dx\,dz, \quad dx\,dy\,dz, \quad dx\,dz\,dy$

Different orders produce different-looking iterated integrals, but they all evaluate to the same number (provided Fubini's theorem applies). The choice matters because some orders lead to much simpler computations than others.

For example, integrating over the tetrahedron bounded by $x + y + z = 1$ and the coordinate planes can be written as:

$\int_0^1 \int_0^{1-x} \int_0^{1-x-y} f(x,y,z) \, dz \, dy \, dx \quad \text{or} \quad \int_0^1 \int_0^{1-z} \int_0^{1-y-z} f(x,y,z) \, dx \, dy \, dz$

These represent the same integral. The key skill is figuring out which order makes the bounds (and the resulting integrals) simplest.

Fubini's Theorem

Fubini's theorem guarantees that if $f(x, y, z)$ is continuous on the domain $D$, then the triple integral's value does not depend on the order of integration. All six orderings yield the same result.

This is what gives you the freedom to pick whichever order makes the computation easiest. Without Fubini's theorem, you'd have no reason to believe that switching the order of integration is valid.

Properties of Triple Integrals

These properties mirror what you saw with double integrals and single integrals. They're useful for simplifying computations and breaking hard problems into manageable pieces.

Linearity Property

For constants $\alpha$ and $\beta$ and integrable functions $f$ and $g$ on $D$:

$\iiint_D [\alpha f(x,y,z) + \beta g(x,y,z)] \, dV = \alpha \iiint_D f(x,y,z) \, dV + \beta \iiint_D g(x,y,z) \, dV$

This says two things at once:

• Sums split apart. You can integrate a sum term by term.
• Constants pull out. Scalar multiples can be factored out of the integral.

For example, if you need $\iiint_D [3(x^2 + y^2 + z^2) - 2xyz] \, dV$, you can compute $\iiint_D (x^2 + y^2 + z^2) \, dV$ and $\iiint_D xyz \, dV$ separately, then combine: multiply the first result by 3, the second by 2, and subtract.

Additivity Property

If $D$ is split into two non-overlapping subdomains $D_1$ and $D_2$ (so $D = D_1 \cup D_2$ with at most shared boundary), then:

$\iiint_D f(x,y,z) \, dV = \iiint_{D_1} f(x,y,z) \, dV + \iiint_{D_2} f(x,y,z) \, dV$

This is useful when the full domain $D$ is awkward to describe with a single set of bounds, but splits naturally into simpler pieces. For instance, an L-shaped solid might be hard to integrate over directly, but you can decompose it into two rectangular boxes, integrate over each, and add the results.

The property extends to any finite number of non-overlapping subdomains, not just two.