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| {{distinguish|Long s|Esh (letter)}}
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| The '''integral symbol''':
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| :<big><big><big>{{Unicode|∫}}</big></big></big> ([[Unicode]]), <math>\displaystyle \int</math> ([[LaTeX]])
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| is used to denote [[integral]]s and [[antiderivatives]] in [[mathematics]]. The notation was introduced by the [[Germany|German]] mathematician [[Gottfried Wilhelm Leibniz]] towards the end of the 17th century. The symbol was based on the {{Unicode|ſ}} ([[long s]]) character, and was chosen because Leibniz thought of the integral as an infinite [[summation|sum]] of infinitesimal [[summand]]s. See [[long s]] for more details on the history of {{Unicode|ſ}}.
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| == Typography in Unicode and LaTeX ==
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| ===Fundamental symbol===
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| {{main|Integral calculus}}
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| The integral symbol is {{unichar|222B|INTEGRAL}} in [[Unicode]]<ref name="Unicode Mathematical Operators">{{cite web| title = Mathematical Operators – Unicode| url = http://www.unicode.org/charts/PDF/U2200.pdf| accessdate = 2013-04-26}}</ref> and <code>\int</code> in [[LaTeX]]. In [[HTML]], it is written as <code>&#x222b;</code> ([[hexadecimal]]), <code>&#8747;</code> ([[decimal]]) and <code>&int;</code> (named "entity").
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| The original [[IBM PC]] [[code page 437]] character set included a couple of characters {{Unicode|⌠}} and {{Unicode|⌡}} (codes 244 and 245, respectively) to build the integral symbol. These were deprecated in subsequent [[MS-DOS]] [[code page]]s, but they still remain in [[Unicode]] (U+2320 and U+2321, respectively) for compatibility.
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| The {{Unicode|∫}} symbol is very similar to, but not to be confused with, the [[ʃ]] symbol (called "[[esh (letter)|esh]]").
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| ===Extensions of the symbol===
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| {{see also|Multiple integral}}
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| Related symbols include:<ref name="Unicode Mathematical Operators"/><ref name="Unicode Supplemental Mathematical Operators">{{cite web| title = Supplemental Mathematical Operators – Unicode| url = http://www.unicode.org/charts/PDF/U2A00.pdf| accessdate = 2013-05-05}}</ref>
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| :{| class="wikitable" | |
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| ! Meaning
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| ! scope="col" colspan="2"| [[Unicode]]
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| ! scope="col" colspan="2"| [[LaTeX]]
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| |-
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| ! [[Double integral]]
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| |style="font-size:20pt;"| {{Unicode|∬}}
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| || U+222C
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| || <math>\iint</math>
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| || <code>\iint</code>
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| ! [[Triple integral]]
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| |style="font-size:20pt;"| {{Unicode|∭}}
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| || U+222D
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| || <math>\iiint</math>
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| || <code>\iiint</code>
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| |-
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| ! [[Contour integral]]
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| |style="font-size:20pt;"| {{Unicode|∮}}
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| || U+222E
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| || <math>\oint</math>
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| || <code>\oint</code>
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| ! [[Clockwise]] integral
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| |style="font-size:20pt;"| {{Unicode|∱}}
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| || U+2231
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| ! [[Anticlockwise]] integral
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| |style="font-size:20pt;"| {{Unicode|⨑}}
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| || U+2A11
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| |-
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| ! Clockwise contour integral
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| |style="font-size:20pt;"| {{Unicode|∲}}
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| || U+2232
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| || {{Intorient|symbol=varoint}}
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| || <code>\varointclockwise</code>
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| ! Anticlockwise contour integral
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| |style="font-size:20pt;"| {{Unicode|∳}}
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| || U+2233
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| || {{Intorient|symbol=ointctr}}
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| || <code>\ointctrclockwise</code>
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| |-
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| ! [[closed manifold|Closed]] [[surface integral]]
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| |style="font-size:20pt;"| {{Unicode|∯}}
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| || U+222F
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| || {{oiint}}
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| || <code>\oiint</code>
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| ! Closed [[volume integral]]
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| |style="font-size:20pt;"| {{Unicode|∰}}
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| || U+2230
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| || {{oiiint}}
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| || <code>\oiiint</code>
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| |-
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| |}
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| == Typography in other languages ==
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| [[File:Integral Uprightness.svg|150px|thumb|Regional variations (English, German, Russian) of the integral symbol.]]
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| In other languages, the shape of the integral symbol differs slightly from the shape commonly seen in English-language textbooks. While the [[English language|English]] integral symbol leans to the right, the [[German language|German]] symbol (used throughout [[Central Europe]]) is upright, and the [[Russian language|Russian]] variant leans to the left.
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| Another difference is in the placement of [[limits of integration|limits]] for [[definite integral]]s. Generally, in English-language books, limits go to the right of the integral symbol: <math> \int_0^T f(t)\;dt </math>.
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| By contrast, in German and Russian texts, [[limits of integration|limits]] for [[definite integral]]s are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing: <math> \int\limits_0^T f(t)\;\mathrm{d}t </math>.
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| == See also ==
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| * [[Capital sigma notation]]
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| * [[Capital pi notation]]
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| == Notes ==
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| {{Reflist}}
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| == References==
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| *{{cite book | last = Stewart | first = James | title = Single Variable Calculus: Early Transcendentals | accessdate = 2006-06-26 | edition = 5th edition | year = 2003 | publisher = Brooks/Cole | location = [[Belmont, California|Belmont, CA]] | isbn = 0-534-39330-6 | chapter = Integrals | page = 381}}
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| *{{citation | last1 = Zaitcev | first1 = V. | last2 = Janishewsky | first2 = A. | last3 = Berdnikov | first3 = A. | contribution = Russian Typographical Traditions in Mathematical Literature | contribution-url = http://www.uni-giessen.de/partosch/eurotex99/zaitsev.pdf | series = EuroTeX'99 Proceedings | year = 1999}}
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| == External links ==
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| * [http://www.fileformat.info/info/unicode/char/222b/index.htm Fileformat.info]
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| {{Infinitesimals}}
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| [[Category:History of calculus]]
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| [[Category:Mathematical symbols]]
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