Riemann sum

A finite sum of function values multiplied by subinterval widths, used to approximate a definite integral. Its value depends on the partition and on where each function value is sampled.

Partition of an interval: A finite division of an interval into adjacent subintervals. The subinterval widths in a Riemann sum come from this division.
Definite integral: A signed accumulation of a function over an interval, defined through limits of sums. Riemann sums provide the limiting construction of this quantity.
Numerical integration: The computation of approximate values for integrals using finite numerical procedures. Riemann sums are among the simplest methods for numerical integration.
Upper and lower sums: Bounds formed by using each subinterval’s supremum or infimum as its representative function value. They bracket ordinary Riemann sums and help characterize integrability.
Sample point: A chosen point in a subinterval at which a function is evaluated. Each term uses a function value sampled from one subinterval.
Limit of a sequence: A value that the terms of a sequence approach as their index increases. Refining partitions produces a sequence of sums whose limit may be the integral.
Error analysis: The study of bounds and estimates for errors in calculations or approximations. The partition size and sampling rule determine how far a sum may be from the integral.
Darboux integral: An integral defined by the agreement of upper and lower sums as partitions are refined. It gives an equivalent formulation of Riemann integrability without arbitrary sample points.
Left Riemann sum: A Riemann sum that samples the function at each subinterval’s left endpoint. This endpoint rule gives one standard choice of sample points.
Continuity (mathematics): A property of a function whose values change without jumps at each point in its domain. Continuous functions on closed bounded intervals are Riemann integrable.

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