Integral symbol

∫
∫
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

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, archived from the original on December 27, 2016, 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, archived from the original on December 27, 2016, 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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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