The integral sign $$\int$$ is a stylised version of the long s letter.

The long s is a typographic variant of lowercase s, being the only lowercase s in the Carolingian minuscule script. The modern short (round) s appeared later to the ends of words, and has now replaced completely the long s in the antiqua script.

Gottfried Wilhelm Leibniz introduced the integral sign as the first letter s of the Latin word summa (`sum'). The long shape of $\displaystyle\int$ may be thought to symbolically depict the fact that integral is a limiting case of sum.

A variant $$\oint$$ of the integral sign is used in integrals taken along a closed curve in $\mathbb{R}^2$ or a about a closed surface in $\mathbb{R}^3$ ; see e.g. Cauchy integral theorem, derivation of heat equation.

The function given after the integral sign, i.e. the function to be integrated, is the integrand.