On 29 October 1675 Leibniz was in Paris working on a problem about areas. Until then he had written omn., short for the Latin omnia, “all”, to mean the sum of all the lines of a figure, as the Italian Cavalieri did. Halfway down the page he wrote: It will be useful to write ∫ instead of omn. — Leibniz, manuscript of 29 October 1675 The symbol $\int$ is a long S, for summa, sum. At the time the long s was an ordinary letter in handwriting. It has never been replaced since.
Measuring the area of curved figures is one of the oldest problems in mathematics. Around 250 BC Archimedes showed that the area of a parabolic segment is $\frac43$ of the area of a triangle inscribed in it. It was a stroke of genius, but made to measure: every new curve needed a new idea. By the 17th century, Fermat, Cavalieri, Wallis and others could already compute the area under $y=x^n$ for many values of $n$. What was missing was a general rule. And what was missing was the realization that Newton and Leibniz each reached in their own way: computing areas and computing rates of change are inverse operations.
Working out the area of a rectangle is easy: base times height. A triangle’s is too: half of that. The trouble begins when the figure is curved. There is no way to tile a circle with little squares without leaving or overshooting a piece. For two thousand years mathematicians attacked this problem with the same idea: approximate the curve by straight figures that get smaller and smaller. What changed was the way of writing that idea.