The definite integral

The Calculus was probably first developed by Newton in his "Methodus Calculus" in the year 1671. The problems of Calculus as he stated them were, (i) To find the velocity at any time when the distance is given, and, (ii) to find the distance traversed when the velocity is known. Later in 1687, Newton introduced the method of limits or, as he called it, the method of "prime and ultimate ratios" to substitute for his original method of fluxions. Newton's contemporary, Leibnitz, introduced the Calculus independently of Newton. His development is probably less rigorous, but the notation of Leibnitz is decidedly superior to that of Newton. the Calculus was discovered, not for its own sake, but because in the practical sciences, like Astronomy and Physics, a definite problem required solution, necessitating this addition to the scientists equipment. Consequently greater emphasize was laid on what the Calculus did for them, than on a rigorous treatment of the subject, and not until later, when it was found that the Calculus did not apply to many functions, was the need felt for a more precise definition of the ideas involved in the Calculus