Integral

An integral is a mathematical quantity that represents accumulated values, often as signed area or total change. It can be defined by limits of sums or by more general integration theories.

Riemann integral: An integral defined as the limit of sums of function values over partitions of an interval. It gives the standard limit-of-sums construction for bounded functions on intervals.
Partition of an interval: A finite division of an interval into subintervals. Partitions determine the pieces used in Riemann sums.
Area under a curve: The area between a function’s graph and an axis over a specified interval. A definite integral computes signed area and, under suitable conditions, geometric area.
Indefinite integral: The family of antiderivatives of a function, conventionally written with an arbitrary constant. Unlike a definite integral, it denotes functions rather than a single accumulated value.
Integrability: The property of a function or differential form having a well-defined integral under a specified theory. Different definitions of integral impose different conditions for existence.
Lebesgue integral: An integral defined by measuring sets of function values and extending the construction to broader function classes. It generalizes integration beyond the interval partitions used in Riemann integration.
Limit (mathematics): A value that a function or sequence approaches as its input approaches a specified point or bound. Integral definitions use limits as partitions become arbitrarily fine.
Arc length: The distance measured along a curve between two points. A curve’s length is obtained by integrating its local length element.
Definite integral: A number representing the integral of a function over specified bounds. It is the bounded-interval quantity most commonly associated with accumulated value.
Uniform integrability: A boundedness condition on a family of functions that controls their integral tails. It helps determine when convergence of functions also yields convergence of integrals.

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