Integral symbol
∫ | |
---|---|
Integral symbol | |
In Unicode | U+222B ∫ 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:
is used to denote integrals and antiderivatives in mathematics, especially in calculus.
History
The notation was introduced by the
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
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
| |
Triple integral
|
∭ | U+222D | \iiint
| |
Quadruple integral
|
⨌ | U+2A0C | \iiiint
| |
Contour integral
|
∮ | U+222E | \oint
| |
Clockwise integral | ∱ | U+2231 | ||
Counterclockwise integral
|
⨑ | U+2A11 | ||
Clockwise contour integral | ∲ | U+2232 | \varointclockwise
| |
Counterclockwise contour integral | ∳ | U+2233 | \ointctrclockwise
| |
Closed surface integral | ∯ | U+222F | \oiint
| |
Closed volume integral | ∰ | U+2230 | \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
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:
See also
- Capital sigma notation
- Capital pi notation
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.