Triple Integrals

A triple integral is simply the double integral's big sibling: instead of adding up something over a flat region, you add it up through a solid, like the mass of a part whose material is denser in some places than others. The symbol $\iiint_E f\,dV$ is the same three pieces as before, one dimension up: $E$ (the solid), $dV$ (a tiny box of volume, $dx\,dy\,dz$) and $f$ (how much stuff per unit volume at that point). The nested limits look like a lot, but each one answers a simple question about the shape. Bear with it; it's the same slicing you've been doing.

Setting up. Think about how you'd describe the solid to someone. First, for a point $(x, y)$ on the floor, where does the solid start and stop as you go straight up? That gives the $z$-limits, from the bottom surface to the top surface, and they can depend on $x$ and $y$. Then look down on the solid from above: its shadow on the $xy$-plane is the region for the remaining double integral, which you set up exactly as in the last few lessons. So a triple integral is really a double integral of a column total: $\iint_D\left(\int_{h_1}^{h_2} f\,dz\right)dA$.

What it computes:

• $\iiint_E 1\,dV$ is the volume of $E$. (It makes sense: with $f = 1$ the inner integral is $h_2 - h_1$, the column's height, and adding heights over the shadow is the volume method from the double-integral lessons.)
• $\iiint_E \rho\,dV$ is the mass of a solid with density $\rho(x, y, z)$.
• Dividing $\iiint x\rho\,dV$ by the mass gives the center of mass's $x$-coordinate: a density-weighted average of $x$.

A common mistake. It's tempting to put the solid's highest and lowest $z$-values on the inner integral, like $0 \le z \le 1$ for the tetrahedron below. Those are only right for a box. Over most points of the shadow the solid stops lower, at the slanted face $z = 1 - x - y$, and the limits have to say so.

There are six possible orders of integration; pick the one with the simplest limits. Slicing at fixed heights, as in the animation below, is another way to set up the same integral.

Mass, center of mass and moments of inertia of real parts (engine blocks, aircraft wings, prosthetics) are triple integrals of density, computed by CAD software.