Integral symbol

The integral symbol:

\displaystyle \int

(LaTeX)

is used to denote integrals and antiderivatives in mathematics.

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 paper "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 ʃ symbol (called "esh").

Extensions of the symbol

Related symbols include:^[5]^[6]^[7]

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

Typography in other languages

Regional variations (English, German, Russian) 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 integral symbol leans to the right, the German symbol (used throughout Central Europe) is upright, and the Russian variant leans to the left.

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}^{T}f(t)\;\mathrm {d} t

By contrast, in German and Russian texts, limits for definite integrals are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing:

\int\limits_0^T 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 ("Analyseos tetragonisticae pars secunda", October 29, 1675) and 321–331 ("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.
1 2 "Mathematical Operators – Unicode" (PDF). Retrieved 2013-04-26.
↑ "Supplemental Mathematical Operators – Unicode" (PDF). Retrieved 2013-05-05.

↑ In the current font, ∫ doesn't have the same style as the others, although on (Mathematical Operators – Unicode), with the font they are using, all the other integrals have the same nice style as ∫. Furthermore ∫ doesn't have the same vertical alignment.

$\int$	$\iint$	$∭$	$\oint$
`int`	`iint`	`iiint`	`oint`

$∲$	$∳$	$∯$	$∰$
`varointclockwise`	`ointctrclockwise`	`oiint`	`oiiint`

NB:

Applying font-style: italic; or font-style: oblique; to the int symbol ∫ has no effect in Firefox or Safari, it remains upright (whereas it does have an effect on all the other integral symbols), e.g.
<span style="font-style: italic;">∫ ∬ ∭ ∮ ∲ ∳ ∯ ∰</span> yields ∫ ∬ ∭ ∮ ∲ ∳ ∯ ∰;

<span style="font-style: oblique;">∫ ∬ ∭ ∮ ∲ ∳ ∯ ∰</span> yields ∫ ∬ ∭ ∮ ∲ ∳ ∯ ∰.

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. (1999), "Russian Typographical Traditions in Mathematical Literature" (PDF), Russian Typographical Traditions in Mathematical Literature, EuroTeX'99 Proceedings

External links

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Infinitesimals
History	Adequality Leibniz's notation Integral symbol Criticism of non-standard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle Method of indivisibles
Related branches	Non-standard analysis Non-standard calculus Internal set theory Synthetic differential geometry Constructive non-standard 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
Scientists	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler
Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

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