Integral symbol: Difference between revisions

∫
∫
Integral symbol
In Unicode	U+222B ∫ INTEGRAL (∫, &Integral;)
Graphical variants
Different from
Different from	U+017F ſ LATIN SMALL LETTER LONG S; U+0283 ʃ LATIN SMALL LETTER ESH

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The integral symbol:

∫ (Unicode),

\displaystyle \int

(LaTeX)

is used to denote integrals and antiderivatives in mathematics, especially in calculus.

History

The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1675 in his private writings;^[1]^[2] it first appeared publicly in the article "De Geometria Recondita et analysi indivisibilium atque infinitorum" (On a hidden geometry and analysis of indivisibles and infinites), published in Acta Eruditorum in June 1686.^[3]^[4] The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands.

Typography in Unicode and LaTeX

Fundamental symbol

The integral symbol is U+222B ∫ INTEGRAL in Unicode^[5] and \int in LaTeX. In HTML, it is written as ∫ (hexadecimal), ∫ (decimal) and ∫ (named entity).

The original IBM PC code page 437 character set included a couple of characters ⌠ and ⌡ (codes 244 and 245 respectively) to build the integral symbol. These were deprecated in subsequent MS-DOS code pages, but they still remain in Unicode (U+2320 and U+2321 respectively) for compatibility.

The ∫ symbol is very similar to, but not to be confused with, the letter ʃ ("esh").

Extensions of the symbol

Related symbols include:^[5]^[6]

Meaning	Unicode		LaTeX
Double integral	∬	U+222C	$\iint$	`\iint`
Triple integral	∭	U+222D	$\iiint$	`\iiint`
Quadruple integral	⨌	U+2A0C	$\iiiint$	`\iiiint`
Contour integral	∮	U+222E	$\oint$	`\oint`
Clockwise integral	∱	U+2231
Counterclockwise integral	⨑	U+2A11
Clockwise contour integral	∲	U+2232	$\varointclockwise$	`\varointclockwise`
Counterclockwise contour integral	∳	U+2233	$\ointctrclockwise$	`\ointctrclockwise`
Closed surface integral	∯	U+222F	$\oiint$	`\oiint`
Closed volume integral	∰	U+2230	$\oiiint$	`\oiiint`

Typography in other languages

In other languages, the shape of the integral symbol differs slightly from the shape commonly seen in English-language textbooks. While the English integral symbol leans to the right, the German symbol (used throughout Central Europe) is upright, and the Russian variant leans slightly to the left to occupy less horizontal space.^[7]

Another difference is in the placement of limits for definite integrals. Generally, in English-language books, limits go to the right of the integral symbol:

$\int _{0}^{5}f(t)\,\mathrm {d} t,\quad \int _{g(t)=a}^{g(t)=b}f(t)\,\mathrm {d} t.$

By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing but is more compact horizontally, especially when using longer expressions in the limits:

$\int \limits _{0}^{T}f(t)\,\mathrm {d} t,\quad \int \limits _{\!\!\!\!\!g(t)=a\!\!\!\!\!}^{\!\!\!\!\!g(t)=b\!\!\!\!\!}f(t)\,\mathrm {d} t.$

Notes

^ Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674–1676, Berlin: Akademie Verlag, 2008, pp. 288–295 Archived 2021-10-09 at the Wayback Machine ("Analyseos tetragonisticae pars secunda", October 29, 1675) and 321–331 Archived 2016-10-03 at the Wayback Machine ("Methodi tangentium inversae exempla", November 11, 1675).
^ Aldrich, John. "Earliest Uses of Symbols of Calculus". Retrieved 20 April 2017.
^ Swetz, Frank J., Mathematical Treasure: Leibniz's Papers on Calculus – Integral Calculus, Convergence, Mathematical Association of America, retrieved February 11, 2017
^ Stillwell, John (1989). Mathematics and its History. Springer. p. 110.
^ ^a ^b "Mathematical Operators – Unicode" (PDF). Retrieved 2013-04-26.
^ "Supplemental Mathematical Operators – Unicode" (PDF). Retrieved 2013-05-05.
^ "Russian Typographical Traditions in Mathematical Literature" (PDF). giftbot.toolforge.org. Archived from the original (PDF) on 28 September 2012. Retrieved 11 October 2021.

References

Stewart, James (2003). "Integrals". Single Variable Calculus: Early Transcendentals (5th ed.). Belmont, CA: Brooks/Cole. p. 381. ISBN 0-534-39330-6.
Zaitcev, V.; Janishewsky, A.; Berdnikov, A. (20 September 1999). Russian Typographical Traditions in Mathematical Literature (PDF). EuroTeX'99. EuroTeX'99 Proceedings. Heidelberg. Archived from the original (PDF) on 24 February 2017.

External links

Fileformat.info

[1] Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674–1676, Berlin: Akademie Verlag, 2008, pp. 288–295 Archived 2021-10-09 at the Wayback Machine ("Analyseos tetragonisticae pars secunda", October 29, 1675) and 321–331 Archived 2016-10-03 at the Wayback Machine ("Methodi tangentium inversae exempla", November 11, 1675).

[2] Aldrich, John. "Earliest Uses of Symbols of Calculus". Retrieved 20 April 2017.

[3] Swetz, Frank J., Mathematical Treasure: Leibniz's Papers on Calculus – Integral Calculus, Convergence, Mathematical Association of America, retrieved February 11, 2017

[4] Stillwell, John (1989). Mathematics and its History. Springer. p. 110.

[Unicode_Mathematical_Operators-5] "Mathematical Operators – Unicode" (PDF). Retrieved 2013-04-26.

[Unicode_Supplemental_Mathematical_Operators-6] "Supplemental Mathematical Operators – Unicode" (PDF). Retrieved 2013-05-05.

[7] "Russian Typographical Traditions in Mathematical Literature" (PDF). giftbot.toolforge.org. Archived from the original (PDF) on 28 September 2012. Retrieved 11 October 2021.

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[5]

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[7]

@@ Line 4: / Line 4: @@
 |sign=∫
 |name=Integral symbol
-|unicode={{unichar|222B|Integral symbol|html=}}
+|unicode={{unichar|222B|Integral|html=}}
 |variant1=<math>\displaystyle \int</math>
 |different from={{unichar|017F|Long s|nlink=Long s}}<br>{{unichar|0283|Esh|nlink=Esh (letter)}}
@@ Line 10: / Line 10: @@
 The '''integral symbol''':
-:<span style=font-size:200%>∫</span> ([[Unicode]]), <math>\displaystyle \int</math> ([[LaTeX]])<br/>
+{{block indent | em = 1.5 | text = <span style=font-size:200%>∫</span> ([[Unicode]]), <math>\displaystyle \int</math> ([[LaTeX]])}}
 is used to denote [[integral]]s and [[Antiderivative|antiderivatives]] in [[mathematics]], especially in [[calculus]].
@@ Line 23: / Line 23: @@
 {{main|Integral calculus}}
-The integral symbol is {{unichar|222B|INTEGRAL}} in [[Unicode]]<ref name="Unicode Mathematical Operators">{{cite web| title = Mathematical Operators – Unicode| url = https://www.unicode.org/charts/PDF/U2200.pdf| access-date = 2013-04-26}}</ref> and <code>\int</code> in [[LaTeX]]. In [[HTML]], it is written as  <code>&amp;#x222b;</code> ([[hexadecimal]]), <code>&amp;#8747;</code> ([[decimal]]) and  <code>&amp;int;</code> ([[List_of_XML_and_HTML_character_entity_references#Character_entity_references_in_HTML|named entity]]).
+The integral symbol is {{unichar|222B|INTEGRAL}} in [[Unicode]]<ref name="Unicode Mathematical Operators">{{cite web| title = Mathematical Operators – Unicode| url = https://www.unicode.org/charts/PDF/U2200.pdf| access-date = 2013-04-26}}</ref> and <code>\int</code> in [[LaTeX]]. In [[HTML]], it is written as  <code>&amp;#x222b;</code> ([[hexadecimal]]), <code>&amp;#8747;</code> ([[decimal]]) and  <code>&amp;int;</code> ([[List of XML and HTML character entity references#Character entity references in HTML|named entity]]).
 The original [[IBM PC]] [[code page 437]] character set included a couple of characters ⌠ and ⌡ (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]] ([[Miscellaneous Technical#(2300–233F)|U+2320 and U+2321]] respectively) for compatibility.
@@ Line 34: / Line 34: @@
 Related symbols include:<ref name="Unicode Mathematical Operators"/><ref name="Unicode Supplemental Mathematical Operators">{{cite web| title = Supplemental Mathematical Operators – Unicode| url = https://www.unicode.org/charts/PDF/U2A00.pdf| access-date = 2013-05-05}}</ref>
-:{| class="wikitable"
+{| class="wikitable"
 |-
 ! Meaning
@@ Line 52: / Line 52: @@
 || <code>\iiint</code>
 |-
-![[Quadruple integral]]
+! [[Quadruple integral]]
 |style="font-size:20pt;"| ⨌
 || U+2A0C
 || <math>\iiiint</math>
 || <code>\iiiint</code>
 |-
 ! [[Contour integral]]
 |style="font-size:20pt;"| ∮
 || U+222E
@@ Line 100: / Line 99: @@
 || {{oiiint}}
 || <code>\oiiint</code>
-|-
 |}
 == Typography in other languages ==
-[[File:Integral Uprightness.svg|150px|thumb|Regional variations (English, German, Russian) of the integral symbol]]
+[[File:Integral Uprightness.svg|150px|thumb|Regional variations ([[English language|English]], [[German language|German]], and [[Russian language|Russian]] from left to right) of the integral symbol]]
 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 slightly to the left to occupy less horizontal space.<ref>{{cite web |title=Russian Typographical Traditions in Mathematical Literature |url=https://giftbot.toolforge.org/deref.fcgi?url=https%3A%2F%2F1.800.gay%3A443%2Fhttp%2Fwww.uni-giessen.de%2Fpartosch%2Feurotex99%2Fzaitsev.pdf |publisher=giftbot.toolforge.org |access-date=11 October 2021 |archive-url=https://web.archive.org/web/20120928083018/http://www.uni-giessen.de/partosch/eurotex99/zaitsev.pdf |archive-date=28 September 2012}}</ref>
 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>
+<math display="block">
  \int_0^5 f(t) \,\mathrm{d}t, \quad
  \int_{g(t) = a}^{g(t) = b} f(t) \,\mathrm{d}t.
 </math>
-By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing, but is more compact horizontally, especially when longer expressions are used in the limits:
+By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing but is more compact horizontally, especially when using longer expressions in the limits:
-: <math>
+<math display="block">
  \int\limits_0^T f(t) \,\mathrm{d}t, \quad
  \int\limits_{\!\!\!\!\!g(t) = a\!\!\!\!\!}^{\!\!\!\!\!g(t) = b\!\!\!\!\!} f(t) \,\mathrm{d}t.
@@ Line 130: / Line 128: @@
 == References==
-*{{cite book | last = Stewart | first = James | title = Single Variable Calculus: Early Transcendentals | edition = 5th | year = 2003 | publisher = Brooks/Cole | location = [[Belmont, California|Belmont, CA]] | isbn = 0-534-39330-6 | chapter = Integrals | page = [https://archive.org/details/singlevariableca00stew/page/381 381] | chapter-url-access = registration | chapter-url = https://archive.org/details/singlevariableca00stew/page/381 }}
+* {{Cite book |last=Stewart |first=James |year=2003 |chapter=Integrals |chapter-url-access=registration |chapter-url=https://archive.org/details/singlevariableca00stew/page/381 |title=Single Variable Calculus: Early Transcendentals |edition=5th |location=Belmont, CA |publisher=Brooks/Cole |isbn=0-534-39330-6 |page=[https://archive.org/details/singlevariableca00stew/page/381 381] }}
-*{{citation | last1 = Zaitcev | first1 = V. | last2 = Janishewsky | first2 = A. | last3 = Berdnikov | first3 = A. | title = Russian Typographical Traditions in Mathematical Literature | contribution = Russian Typographical Traditions in Mathematical Literature | contribution-url = https://www.staff.uni-giessen.de/~g029/eurotex99/zaitsev.pdf | series = EuroTeX'99 Proceedings | year = 1999}}
+* {{Cite conference |last1=Zaitcev |first1=V. |last2=Janishewsky |first2=A. |last3=Berdnikov |first3=A. |date=20 September 1999 |title=Russian Typographical Traditions in Mathematical Literature |url=https://www.staff.uni-giessen.de/~g029/eurotex99/zaitsev.pdf |archive-url=https://web.archive.org/web/20170224213730/https://www.staff.uni-giessen.de/~g029/eurotex99/zaitsev.pdf |archive-date=24 February 2017 |url-status=dead |location=Heidelberg |conference=EuroTeX'99 |work=EuroTeX'99 Proceedings}}
 == External links ==

v t e Infinitesimals
History	Adequality Leibniz's notation Integral symbol Criticism of nonstandard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle
Related branches	Nonstandard analysis Nonstandard calculus Internal set theory Synthetic differential geometry Smooth infinitesimal analysis Constructive nonstandard analysis Infinitesimal strain theory (physics)
Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers
Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental law of homogeneity
Mathematicians	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler
Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse