Iterated integral

An integral formed by applying integration repeatedly, often as nested integrals over multiple variables. Its value can depend on the order of integration when domains or integrands require care.

Fubini's theorem: A theorem giving conditions under which a multiple integral equals either order of iterated integration. It justifies exchanging the order of integration when its hypotheses hold.
Definite integral: A number representing the signed accumulation of a function over an interval. Each stage of an iterated integral applies a definite integral to one variable.
Area between curves: The area of a planar region bounded by graphs of functions. Integrating slice lengths repeatedly can determine areas of more complex regions.
Line integral: An integral that accumulates a scalar or vector field along a curve. It integrates over a one-dimensional path rather than through nested coordinate ranges.
Tonelli's theorem: A theorem equating repeated and multiple integrals for nonnegative measurable functions, including when the value is infinite. Nonnegativity permits order exchange without assuming integrability in advance.
Multiple integral: An integral that accumulates a function over a region in two or more dimensions. Under suitable conditions, iterated integration computes a multiple integral.
Volume of a solid: The three-dimensional measure of the space occupied by a solid. Nested integration sums cross-sectional areas or volume elements.
Surface integral: An integral that accumulates a scalar or vector field over a surface. Its domain is a curved surface, unlike the nested interval bounds typical of iterated integrals.
Change of order of integration: The rewriting of an iterated integral with its integration variables taken in a different order. A reversed order can simplify limits or make an integral evaluable.
Integration region: The set of points over which an integral accumulates a function. Its geometry determines the bounds in an equivalent nested integral.

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Cauchy's formula for repeated integrationNarrower topic: The formula's left side is an iterated integral with a fixed lower endpoint.