4.3 Triple Integrals

Understanding Triple Integrals

Triple integrals extend the idea of double integrals into three dimensions. Where a double integral accumulates a quantity over a 2D region, a triple integral accumulates over a 3D solid. They show up whenever you need to compute volumes, masses, or averages across a solid region in space.

Evaluation of Triple Integrals

The simplest case is integrating over a rectangular box where all the limits are constants. The general form looks like:

$\iiint_B f(x,y,z) \, dV = \int_{a}^{b} \int_{c}^{d} \int_{e}^{f} f(x,y,z) \, dz \, dy \, dx$

To evaluate this, you work from the inside out:

1. Integrate with respect to $z$ first, treating $x$ and $y$ as constants.
2. Integrate the result with respect to $y$, treating $x$ as a constant.
3. Integrate the final expression with respect to $x$.

Each step reduces the dimension by one, so after all three integrations you're left with a number.

Quick example: Suppose you want $\int_0^2 \int_0^1 \int_0^3 x^2 y z \, dz \, dy \, dx$. You'd first integrate $x^2 y z$ with respect to $z$ (getting $x^2 y \cdot \frac{z^2}{2}\big|_0^3 = \frac{9}{2}x^2 y$), then integrate that result over $y$, then over $x$.

When $f(x,y,z) = 1$, the triple integral simply gives the volume of the region.

Setup of General Triple Integrals

Most real problems don't involve neat rectangular boxes. The region might be bounded by spheres, cylinders, cones, or planes. Setting up the integral correctly is usually the hardest part. Here's a reliable process:

1. Sketch the region. Even a rough sketch helps you see which surfaces form the top, bottom, and sides of the solid.
2. Choose a coordinate system. Cartesian works for regions bounded by planes and simple surfaces. Cylindrical coordinates suit regions with circular symmetry about one axis. Spherical coordinates are best for regions bounded by spheres or cones.
3. Pick an integration order. Decide which variable to integrate first (innermost). A good choice is the variable whose limits depend on the other two, leaving the outermost variable with constant limits.
4. Express the limits. For the innermost variable, the limits are generally functions of the other two variables. For the middle variable, limits are functions of the outermost variable. The outermost variable has constant limits.
5. Write the integral and evaluate.

For non-rectangular regions, the integral takes a form like:

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

Notice how the $z$-limits can depend on both $x$ and $y$, and the $y$-limits can depend on $x$. Getting these dependency relationships right is the core challenge.

Applications of Triple Integrals

Volume of a solid region $R$:

$V = \iiint_R 1 \, dV$

This is the 3D analog of finding area with a double integral.

Mass of a solid with density function $\rho(x,y,z)$:

$M = \iiint_R \rho(x,y,z) \, dV$

The density function tells you how mass is distributed throughout the solid. If $\rho$ is constant, mass is just density times volume.

Center of mass requires computing three coordinates $(\bar{x}, \bar{y}, \bar{z})$, each found by dividing a first moment by the total mass:

• $\bar{x} = \frac{1}{M} \iiint_R x\,\rho(x,y,z) \, dV$
• $\bar{y} = \frac{1}{M} \iiint_R y\,\rho(x,y,z) \, dV$
• $\bar{z} = \frac{1}{M} \iiint_R z\,\rho(x,y,z) \, dV$

Each moment integral weights position by density. The center of mass is the "balance point" of the solid. For uniform density, it reduces to the centroid, which depends only on geometry.

Order of Integration

For a rectangular box, all six possible orders of integration give the same answer (by Fubini's theorem). But for general regions, some orders are far easier to set up and compute than others. When choosing an order:

• Look for constant limits. If one variable ranges between two fixed numbers regardless of the others, make it the outermost integral.
• Check the integrand. Sometimes one order lets you integrate a complicated function early, simplifying what remains.
• Consider the region's geometry. If the solid is easiest to describe by saying "for each $(x,y)$, $z$ goes from one surface to another," then $z$ should be the innermost variable.
• Watch for symmetry. If the region and integrand are symmetric about an axis, you can sometimes conclude that certain integrals are zero without computing them.

A common mistake is writing limits that don't actually describe the intended region. After setting up your integral, it's worth checking a few boundary points to confirm the limits are correct before you start computing.