The integral symbol:
The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1675 in his private writings; 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. 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.
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.
|Clockwise contour integral||∲||U+2232|
|Counterclockwise contour integral||∳||U+2233|
|Closed surface integral||∯||U+222F|
|Closed volume integral||∰||U+2230|
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.
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: