triple integral

A triple integral is the double integral pushed one more dimension up: instead of summing a quantity over a flat region, you sum it over a solid lump of three-dimensional space. You cannot picture the volume under a four-dimensional graph, so it is cleaner to think of it directly: a solid body E sits in space, and at each point of it there is some density f(x, y, z) — mass per unit volume, charge per unit volume, or just the number 1. The triple integral of f over E adds up f times a tiny chunk of volume over the whole body and reports the grand total.

The construction mirrors the double integral exactly. Carve E into many small boxes of volume delta-V = delta-x times delta-y times delta-z, evaluate f at a sample point in each box, form the products f delta-V, and add. The limit as the boxes shrink is written as the triple integral over E of f(x, y, z) dV, where the volume element dV in Cartesian coordinates is dx dy dz. When f is the constant 1, the triple integral simply returns the volume of E. As with double integrals you evaluate it as three nested ordinary integrals, peeling off one variable at a time, with the inner limits describing how the solid's cross-section changes as you move outward.

Triple integrals are the natural language for everything a solid body carries: its total mass from a variable density rho(x, y, z), its center of mass, its moments of inertia about an axis, the total gravitational or electrostatic energy stored in a region, or the probability mass of three jointly distributed random variables. In real problems the Cartesian box dx dy dz is rarely convenient — spheres, cylinders, and cones beg for cylindrical or spherical coordinates, whose volume elements (r dr d-theta dz and rho^2 sin(phi) d-rho d-phi d-theta) absorb the geometry of the region and make the limits simple.