Multiple integral

A multiple integral accumulates values of a function over a domain with two or more dimensions. Double and triple integrals are its most common forms.

Fubini's theorem: A theorem that permits an iterated integral to equal a multiple integral under suitable conditions. It turns many double and triple integrals into successive one-variable integrations.
Multivariable function: A function whose input has two or more coordinates. A multiple integral commonly accumulates the values of such a function across a domain.
Mass: A measure of the amount of matter in an object. Integrating a density over a region gives the total mass of a body.
Single-variable integral: An integral that accumulates a function over a one-dimensional interval. It is the one-dimensional counterpart, with interval length replacing multidimensional volume.
Change of variables: A method for transforming integrals by replacing variables and accounting for how the transformation changes volume. It reshapes a difficult domain and adjusts the integrand through a Jacobian determinant.
Domain of integration: The set of input points over which an integral is evaluated. Its shape and bounds determine which function values contribute.
Center of mass: The point representing the average position of an object's mass. Its coordinates come from integrating position weighted by density.
Line integral: An integral of a function or vector field along a curve. It accumulates along a one-dimensional path rather than throughout a multidimensional region.
Jacobian matrix: A matrix of first derivatives describing the local linear behavior of a multivariable function. Its determinant supplies the local area or volume scaling in coordinate transformations.
Area: A measure of the two-dimensional extent of a region. A double integral of the constant function one computes a region's area.

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