# Integral symbol

is used to denote integrals and antiderivatives in mathematics.

The **integral symbol**:

## 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`

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 slight to the left to occupy less horizontal space.

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:

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:

## 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. - "Mathematical Operators – Unicode" (PDF). Retrieved 2013-04-26.
- "Supplemental Mathematical Operators – Unicode" (PDF). Retrieved 2013-05-05.

## 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