Definite integral

A signed accumulation of a function over an interval, defined as the limit of sums of function values multiplied by subinterval widths. It measures net area and extends to many forms of accumulated quantity.

Riemann sum: A finite sum of function values multiplied by subinterval widths, used to approximate an integral. Its limit defines the definite integral for Riemann-integrable functions.
Function: A rule assigning each input in a domain exactly one output. The integrand supplies the values whose accumulation the integral measures.
Fundamental theorem of calculus: The theorem connecting definite integrals with antiderivatives and derivatives. It often evaluates an integral through endpoint values of an antiderivative.
Lebesgue integral: An integral defined by measuring sets of function values rather than partitioning the domain into intervals. It integrates broader classes of functions than the Riemann definition.
Partition of an interval: A finite subdivision of an interval into smaller intervals. Refining partitions makes the sums capture the function's accumulation more precisely.
Interval: A set of real numbers containing every number between any two of its members. The endpoints specify the domain over which accumulation is taken.
Net change theorem: A result stating that integrating a rate over time gives the change in the measured quantity. It interprets signed accumulation as total change between interval endpoints.
Riemann–Stieltjes integral: An integral formed from sums weighted by increments of a function called the integrator. It generalizes interval-width weights in Riemann sums to other accumulation rules.
Mesh of a partition: The largest subinterval length in a partition. The limiting definition requires partition meshes to shrink toward zero.
Limit: A value approached by a quantity as its input approaches a specified value or process. A limit turns finite approximating sums into an exact integral.

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