Riemann integral

The Riemann integral assigns a value to a bounded function on an interval when sums of sampled function values, weighted by subinterval widths, converge as partitions are refined.

Riemann sum: A finite sum of function values multiplied by the widths of partition subintervals. Its limiting behavior defines the Riemann integral.
Limit: A value that quantities approach as an input or parameter approaches a specified value. The integral is defined by the limit of approximating sums.
Area under a curve: The planar area between a graph and a horizontal axis over an interval. For nonnegative integrable functions, the integral gives this area.
Lebesgue integral: An integral built by measuring sets of function values rather than partitioning the domain. It integrates functions, such as the Dirichlet function, that the Riemann integral cannot.
Lebesgue criterion for Riemann integrability: A bounded function on a closed interval is Riemann integrable exactly when its discontinuities have measure zero. It gives a precise characterization of which bounded functions admit the integral.
Partition of an interval: A finite division of an interval into adjacent subintervals. Partitions specify the widths used to approximate the integral.
Bounded function: A function whose values remain within fixed finite bounds on its domain. Boundedness keeps the sums finite and is part of the standard definition.
Fundamental theorem of calculus: The theorem linking differentiation of antiderivatives with definite integration. It provides a practical way to evaluate many Riemann integrals.
Improper integral: A limit-defined integral for unbounded functions or intervals of infinite length. It extends ordinary integration beyond the bounded-function, finite-interval setting.
Measure zero: A set has measure zero when it can be covered by sets of arbitrarily small total measure. The integrability criterion allows discontinuities only on a measure-zero set.

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