ABSTRACT
When Newton and Leibniz defined the integral, it was as a sum. In fact, the symbol for integral is a stylized letter S (as in “summa,” the Latin for a sum). Soon thereafter, they discovered that the same result can be obtained using antiderivatives. Since their definitions involved the controversial infinitesimals (infinitely small numbers), the mathematical community preferred the idea of the integral as the antiderivative, and in the 18th century that was the prevailing viewpoint. It was only in the 19th century, when the need for a rigorous approach became obvious, that Cauchy and Riemann reverted to the old definition (this time without infinitesimals). Although they made significant progress, there was still room for improvement, which came through the work of Lebesgue in the 20th century.
