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Integral — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Integral
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-08T00:11:09.439Z
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Integral — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Integral (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_409bca135efb (Metraipolis archive dc_409bca135efb, sha256 28d72af4ed95f2f7c67549c342ac2599d84e48a60a43c927338125d881a4f7a1)

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@misc{dc409bca135efb,
  title = {Integral — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Integral},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_409bca135efb; sha256 28d72af4ed95f2f7c67549c342ac2599d84e48a60a43c927338125d881a4f7a1},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Integral. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In mathematics, an integral is the continuous analog of a sum, and is used to calculate areas, volumes, and their generalizations. The process of computing an integral, called integration, is one of the two fundamental operations of calculus, along with differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations. Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting two points in space. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.

Tema degli esploratori: storia e filosofia della scienza (deposito sistematico; ricerca «history of mathematics»).
Fonte: Wikipedia in inglese, voce «Integral» — https://en.wikipedia.org/wiki/Integral
Wikidata Q80091 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q80091
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Integral?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1375764364, consultata il 2026-10-07T23:40:59.780Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 5fa1ebab0613a9c9265ee564851f995aa692e77eb9b62008f4d44bc2023e5958