Triple Integrals

The central idea

For an integrable scalar field $f$ on a solid $E\subset\mathbb R^3$, $\iiint_Ef\,dV$ is the limit of weighted volume sums. With $f=1$ it gives volume; with density it gives mass. Fubini's theorem permits iterated integration when appropriate integrability hypotheses hold, and bounds must describe each point of $E$ exactly once.

Definitions, hypotheses, and notation

A solid can be simple in one order and require several pieces in another. Start from the innermost coordinate as a line segment between two surfaces, then project onto a planar region already understood from double integrals. In Cartesian coordinates the Jacobian is one; cylindrical and spherical systems alter it.

Mass moments multiply density by coordinate distances, and centroids divide first moments by total mass. Units provide a strong distinction: density times $dV$ gives mass, while an additional length factor gives a first moment. Symmetry can force some moments to zero before calculation.

Conceptual meaning

A triple integral accumulates through three-dimensional cells. The innermost integral compresses a line of cells, the next combines a sheet, and the outermost completes the solid. Coordinate changes reshape cells and introduce a Jacobian.

A dependable method and decision rule

1. Sketch the solid and its projection onto a coordinate plane.
2. Choose an order and write inner lower/upper surfaces.
3. Describe the projected planar region with the outer bounds.
4. Integrate one variable at a time, holding outer variables fixed.
5. Check volume, units, symmetry, and whether bounds cover the solid once.

Fully worked example

Common mistakes and why they fail