The integral symbol (''see below'') is used to denote

integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

s and antiderivatives in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, especially in

calculus Calculus 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 generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

History

The notation was introduced by the German mathematician

Gottfried Wilhelm Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to ...

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 edit ...

'' 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 Archaism, archaic form of the lowercase letter , found mostly in works from the late 8th to early 19th centuries. It replaced one or both of the letters ''s'' in a double-''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 in

Unicode Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Char ...

and \int in

LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latices are found in nature, but synthetic latices are common as well. In nature, latex is found as a wikt:milky, milky fluid, which is present in 10% of all floweri ...

. In

HTML Hypertext Markup Language (HTML) is the standard markup language for documents designed to be displayed in a web browser. It defines the content and structure of web content. It is often assisted by technologies such as Cascading Style Sheets ( ...

, it is written as ∫ (

hexadecimal Hexadecimal (also known as base-16 or simply hex) is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbo ...

), ∫ (

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 (''decimal fractions'') of th ...

) 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 List of IBM Personal Computer models, IBM PC model line and the basis for the IBM PC compatible ''de facto'' standard. Released on ...

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 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 op ...

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

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 integral symbol leans to the right, the German symbol (used throughout

Central Europe Central Europe is a geographical region of Europe between Eastern Europe, Eastern, Southern Europe, Southern, Western Europe, Western and Northern Europe, Northern Europe. Central Europe is known for its cultural diversity; however, countries in ...

) is upright, and the Russian variant leans slightly 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:

\int_0^5 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 using longer expressions in the limits:

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

Notes

References

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External links

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{{Infinitesimals History of calculus Mathematical symbols Gottfried Wilhelm Leibniz