Triple Integrals

In this lesson you’ll learn how to set up and evaluate triple integrals, which let you accumulate quantities over three-dimensional regions. This is the natural next step after double integrals.

Think about how integration has scaled up through calculus:

• A single integral adds up values along a line (1D). It computes things like area under a curve.
• A double integral adds up values over a flat region (2D). It computes things like volume under a surface, or total mass of a flat plate.
• A triple integral adds up values throughout a solid region (3D). It computes things like total mass of a 3D object, or the volume of a solid.

Each time, you’re doing the same thing: chopping the region into tiny pieces, multiplying the function value by the size of each piece, and adding everything up. The integral is the limit of that sum.

A triple integral over a 3D region E is written as

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

What this computes depends on what f means:

• If f(x, y, z) = 1, the triple integral gives the volume of E. You’re just adding up all the tiny volume elements dV throughout the region.
• If f(x, y, z) is a density (mass per unit volume), the triple integral gives the total mass. Each tiny piece contributes density times volume.
• If f(x, y, z) is a temperature, the triple integral (divided by the volume) gives the average temperature throughout the region.
• In general, the triple integral accumulates whatever f represents over the entire 3D region.

Just like double integrals, you evaluate a triple integral by doing three single integrals, one at a time. In Cartesian coordinates

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

Start with the innermost integral (z), then the middle (y), then the outermost (x). For rectangular boxes, all limits are constants, which makes things simple.

Example 1: Volume of a box

Find the volume of the rectangular box where 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, 0 ≤ z ≤ 4.

With f = 1 (volume), all limits are constants

$$\iiint_E 1 \, dV = \int_0^2 \int_0^3 \int_0^4 dz \, dy \, dx$$

Innermost (z)

$$\int_0^4 dz = 4$$

Middle (y)

$$\int_0^3 4 \, dy = 12$$

Outermost (x)

$$\int_0^2 12 \, dx = 24$$

Volume = 24 cubic units. Which is just 2 x 3 x 4 = 24. The triple integral confirms what geometry already tells us.

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The rectangular box with dimensions 2 x 3 x 4. The triple integral just adds up all the tiny volume elements dV throughout this region, giving 24.

Example 2: Integrating a function over the unit cube

Evaluate the triple integral of f(x, y, z) = x + y + z over the unit cube [0, 1] x [0, 1] x [0, 1].

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The unit cube is the simplest 3D region: all three variables go from 0 to 1 independently.

$$\int_0^1 \int_0^1 \int_0^1 (x + y + z) \, dz \, dy \, dx$$

Innermost (z), treating x and y as constants

$$\int_0^1 (x + y + z) \, dz = \left[ xz + yz + \frac{z^2}{2} \right]_0^1 = x + y + \frac{1}{2}$$

Middle (y), treating x as constant

$$\int_0^1 \left( x + y + \frac{1}{2} \right) dy = \left[ xy + \frac{y^2}{2} + \frac{y}{2} \right]_0^1 = x + 1$$

Outermost (x)

$$\int_0^1 (x + 1) \, dx = \left[ \frac{x^2}{2} + x \right]_0^1 = \frac{1}{2} + 1 = \frac{3}{2}$$

The triple integral equals 3/2. This is the total accumulated value of x + y + z throughout the unit cube.

Example 3: Why the order doesn’t matter for boxes

For rectangular regions, you can integrate in any order. The same integral in dx dz dy order

$$\int_0^1 \int_0^1 \int_0^1 (x + y + z) \, dx \, dz \, dy$$

gives the same answer: 3/2. This is the 3D version of Fubini’s theorem. For non-rectangular regions, the order still doesn’t change the answer, but it does change the limits, and some orders are easier than others.

Triple integrals are the tool for working with 3D quantities:

• Physics uses them to compute total mass, center of mass, and moments of inertia for solid objects with varying density
• Engineering uses them for heat distribution through 3D materials, fluid flow through pipes, and stress analysis in solid structures
• Game engines approximate triple integrals when computing volumetric effects like fog density, 3D lighting, or the total mass of a physics object for realistic simulation
• Medical imaging uses them to analyze 3D scan data from CT and MRI, computing total tissue volume or average density in a region