Integral calculus

Integral calculus studies integration: how quantities accumulate over intervals and regions, and how integration recovers functions from their derivatives.

Definite integral: A number representing the signed accumulation of a function over an interval. It formalizes accumulation across an interval, the central quantitative operation of integral calculus.
Limit (mathematics): A value that a function or sequence approaches as its input approaches a specified point or grows without bound. Limits make precise the sums and convergence on which integration depends.
Area: A measure of the two-dimensional extent of a plane region. Definite integrals calculate areas under curves and between graphs.
Differential calculus: The branch of calculus concerned with derivatives, rates of change, and local behavior. It studies the operation that integration reverses under the fundamental theorem.
Archimedes: An ancient Greek mathematician and engineer known for results in geometry, mechanics, and hydrostatics. His method of exhaustion anticipated integral techniques for finding areas and volumes.
Indefinite integral: The family of antiderivatives of a function, usually written with an arbitrary constant. It expresses the recovery of functions from their derivatives.
Function (mathematics): A rule assigning exactly one output to each input in a specified domain. Integrals aggregate function values across domains or intervals.
Volume: A measure of the three-dimensional space occupied by a solid. Integrals assemble cross-sectional areas or shells to measure solids.
Numerical integration: Methods for approximating integrals with finite computations when exact evaluation is unavailable or impractical. It approximates accumulation rather than expressing the result as an exact antiderivative or limit.
Method of exhaustion: An ancient geometric method that bounds a figure using increasingly precise inscribed or circumscribed figures. It provided a rigorous precursor to calculating areas through limiting approximations.

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