## Integral |

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In **integral** assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of *a*, *b*] of the **definite integral**

is defined informally as the signed *x* = *a* and *x* = *b*. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.

The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. For this reason, the term *integral* may also refer to the related notion of the * indefinite integral* and is written:

The integrals discussed in this article are those termed *definite integrals*. It is the *a*, *b*], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by

The principles of integration were formulated independently by *a*, *b*] is replaced by a

- history
- applications
- terminology and notation
- interpretations of the integral
- formal definitions
- properties
- fundamental theorem of calculus
- extensions
- computation
- see also
- notes
- references
- external links

The first documented systematic technique capable of determining integrals is the *ca.* 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. This method was further developed and employed by

A similar method was independently developed in China around the 3rd century AD by

The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of *x*^{n} up to degree *n* = 9 in

The major advance in integration came in the 17th century with the independent discovery of the

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of

*x*′, which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.

The modern notation for the indefinite integral was introduced by **∫**, from the letter *ſ* (*summa* (written as *ſumma*; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by *Mémoires* of the French Academy around 1819–20, reprinted in his book of 1822 (Cajori 1929, pp. 249–250; Fourier 1822, §231).

Other Languages

አማርኛ: አጠራቃሚ

العربية: تكامل

aragonés: Integración

asturianu: Integración

azərbaycanca: İnteqral

تۆرکجه: انتقرال

বাংলা: সমাকলন

Bân-lâm-gú: Chek-hun

беларуская: Інтэграл

български: Интеграл

bosanski: Integral

català: Integració

Чӑвашла: Интеграл

čeština: Integrál

Cymraeg: Integryn

Deutsch: Integralrechnung

eesti: Integraal

Ελληνικά: Ολοκλήρωμα

español: Integración

Esperanto: Integralo

euskara: Integral

فارسی: انتگرال

français: Intégration (mathématiques)

galego: Integral

客家語/Hak-kâ-ngî: Chit-fûn-ho̍k

한국어: 적분

Հայերեն: Ինտեգրալ

हिन्दी: समाकलन

hrvatski: Integral

Ido: Integralo

Bahasa Indonesia: Integral

íslenska: Heildun

italiano: Integrale

עברית: אינטגרל

ქართული: ინტეგრალი

қазақша: Интеграл

kurdî: Întegral

Latina: Integrale

latviešu: Integrālis

lumbaart: Integral

magyar: Integrál

മലയാളം: സമാകലനം

Malti: L-Integral

मराठी: संकलन

Bahasa Melayu: Kamiran

монгол: Интеграл

မြန်မာဘာသာ: အင်တီဂရေးရှင်း

日本語: 積分法

norsk: Integral (matematikk)

norsk nynorsk: Integral

occitan: Integracion

ភាសាខ្មែរ: អាំងតេក្រាល

polski: Całka

português: Integral

română: Integrală

русский: Интеграл

Scots: Integral

shqip: Integrali

sicilianu: Intiggrali

Simple English: Integral

slovenčina: Integrál

slovenščina: Integral

کوردی: تەواوکاری

српски / srpski: Интеграл

srpskohrvatski / српскохрватски: Integral

Basa Sunda: Integral

suomi: Integraali

svenska: Integral

தமிழ்: தொகையீடு

татарча/tatarça: Интеграл

ไทย: ปริพันธ์

Türkçe: İntegral

українська: Інтеграл

اردو: تکامل

vèneto: Integral

Tiếng Việt: Tích phân

吴语: 定积分

粵語: 積分

Zazaki: İntegral

中文: 积分