Integral symbol (∫)
The integral symbol (∫) marks an integral in calculus. With limits above and below it, it is a definite integral, a number that extends the idea of the area under a curve; without them it is an indefinite integral, a family of functions.
How to read it
In ∫ f(x) dx with a lower limit a and an upper limit b, OpenStax’s Calculus calls a and b the limits of integration, f(x) the integrand and x the variable of integration. The shape is an elongated S, for summa, sum: Gottfried Wilhelm Leibniz chose it, and OpenStax notes that it suggests sigma, the summation sign. The code point is U+222B, INTEGRAL.
Also searched as: ∫, integral sign.
Look-alikes
Unicode cross-refers ∫ to ʃ (U+0283), esh, the phonetic letter for the voiceless postalveolar fricative. Related signs have their own characters: the double integral ∬ (U+222C) and the contour integral ∮ (U+222E), an integral sign with a small circle on it.
Where this symbol came from
oldsymbols.comInfinity symbol (∞)Miller’s history of calculus signs records Wallis’s ∞ (1655) beside Leibniz’s ∫ (1675).More math symbols
Browse symbols drawn with curves and waves.
Sources
- Mathematical Operators: Range 2200–22FF (code chart), The Unicode Standard, Version 18.0, Unicode Consortium, Mathematical Operators, 222B–222E
- IPA Extensions: Range 0250–02AF (code chart), The Unicode Standard, Version 18.0, Unicode Consortium, IPA Extensions, 0283
- Calculus Volume 1, 5.2 The Definite Integral, OpenStax (Rice University), Definition and Notation
- Earliest Uses of Symbols of Calculus, Jeff Miller, Earliest Uses of Various Mathematical Symbols, MacTutor History of Mathematics, University of St Andrews, Integral; Infinity
- A History of Mathematical Notations, Vol. II: Notations Mainly in Higher Mathematics, Florian Cajori (Chicago: Open Court, 1929); full text of the Wellcome Collection copy, Internet Archive, § 539; § 570, p. 203
Last reviewed September 30, 2026 by Rory Hansen. How we research