The ∫ symbol $\int$ is used to denote the integral in mathematics. The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz towards the end of the 17th century. 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. See long s for more details on the history of ſ.
The long s survives in elongated form, and with an italic-style curled descender, as the integral symbol ∫ used in calculus; Gottfried Wilhelm von Leibniz based the character on the Latin word summa ("sum"), which he wrote ſumma.
As for the vertical bar denoting the bounds of evaluating an antiderivative:
This may be called Evaluation Bar. See, in particular, here (Evaluation Bar Notation:).