Volume integral

A volume integral accumulates a scalar or vector quantity over a three-dimensional region. It generalizes area integration to measure totals distributed throughout space.

Triple integral: An iterated integral over three variables that computes accumulation across a three-dimensional region. It is the standard computational form of a volume integral.
Mass density: Mass per unit volume at a point in a material or spatial distribution. Integrating it over a body gives the body's total mass.
Riemann integral: An integral defined as the limit of sums over increasingly fine partitions. Volume integrals extend this sum-and-limit idea to three dimensions.
Line integral: An integral that accumulates a scalar or vector quantity along a curve. It integrates over a one-dimensional path rather than a three-dimensional region.
Volume element: An infinitesimal measure of volume, written in coordinates such as dV or dx dy dz. The integral sums the quantity against this local measure.
Center of mass: The position representing the mass-weighted average location of a system. Volume integrals of density and position determine it for continuous bodies.
Measure theory: The mathematical study of measures, measurable sets, and integration. It formalizes integration over general regions and functions.
Surface integral: An integral that accumulates a quantity across a two-dimensional surface. It measures over a boundary or surface, not through the enclosed volume.
Jacobian determinant: A determinant that describes how a differentiable coordinate transformation scales local volume. It supplies the scaling factor when changing coordinates.
Moment of inertia: A measure of how mass is distributed relative to an axis of rotation. A volume integral weights density by squared distance from the axis.

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