The integral symbol: :∫ (

Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...

\displaystyle \int

(

LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...

)
is used to denote

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

s and antiderivatives in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, especially in

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

History

The notation was introduced by the

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...

mathematician

Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...

in 1675 in his private writings; it first appeared publicly in the article "" (On a hidden geometry and analysis of indivisibles and infinites), published in ''

Acta Eruditorum (from Latin: ''Acts of the Erudite'') was the first scientific journal of the German-speaking lands of Europe, published from 1682 to 1782. History ''Acta Eruditorum'' was founded in 1682 in Leipzig by Otto Mencke, who became its first editor, ...

'' in June 1686. The symbol was based on the ſ (

long s The long s , also known as the medial s or initial s, is an archaic form of the lowercase letter . It replaced the single ''s'', or one or both of the letters ''s'' in a 'double ''s sequence (e.g., "ſinfulneſs" for "sinfulness" and "po� ...

) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal

summand Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...

Typography in Unicode and LaTeX

Fundamental symbol

The integral symbol is in

and \int in

. In

HTML The HyperText Markup Language or HTML is the standard markup language for documents designed to be displayed in a web browser. It can be assisted by technologies such as Cascading Style Sheets (CSS) and scripting languages such as JavaS ...

, it is written as ∫ (

hexadecimal In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, he ...

), ∫ (

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

) and ∫ ( named entity). The original

IBM PC The IBM Personal Computer (model 5150, commonly known as the IBM PC) is the first microcomputer released in the IBM PC model line and the basis for the IBM PC compatible de facto standard. Released on August 12, 1981, it was created by a team ...

code page 437 Code page 437 ( CCSID 437) is the character set of the original IBM PC (personal computer). It is also known as CP437, OEM-US, OEM 437, PC-8, or DOS Latin US. The set includes all printable ASCII characters as well as some accented letters ( d ...

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 MS-DOS ( ; acronym for Microsoft Disk Operating System, also known as Microsoft DOS) is an operating system for x86-based personal computers mostly developed by Microsoft. Collectively, MS-DOS, its rebranding as IBM PC DOS, and a few o ...

code page In computing, a code page is a character encoding and as such it is a specific association of a set of printable characters and control characters with unique numbers. Typically each number represents the binary value in a single byte. (In some c ...

s, but they still remain in

( 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: :

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 English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...

integral symbol leans to the right, the

symbol (used throughout

Central Europe Central Europe is an area of Europe between Western Europe and Eastern Europe, based on a common historical, social and cultural identity. The Thirty Years' War (1618–1648) between Catholicism and Protestantism significantly shaped the a ...

) is upright, and the

Russian Russian(s) refers to anything related to Russia, including: *Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *Rossiyane (), Russian language term for all citizens and peo ...

variant leans slightly to the left to occupy less horizontal space. Another difference is in the placement of

limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

for

definite integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

s. Generally, in English-language books, limits go to the right of the integral symbol: :

\int_0^T f(t) \,\mathrmt, \quad
 \int_^ f(t) \,\mathrmt.

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: :

\int\limits_0^T f(t) \,\mathrmt, \quad
 \int\limits_^ f(t) \,\mathrmt.

Notes

References

* *

External links

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{{Infinitesimals History of calculus Mathematical symbols Gottfried Wilhelm Leibniz