Although calculus-type mathematics was used several times by different mathematicians and scientists throughout history, Leibnitz and Newton are credited with the "invention" of calculus. Developing differentials (the instantaneous slopes of curves) was only a step in the process that finally connected the integral with the derivative and created the system we now know as calculus. However, each piece of this calculus: derivatives, higher order derivatives, definite integrals, indefinite integrals, as well as 2D and 3D versions of all of these, were all used and developed by individuals in the Western world throughout the last 500 years (at least). (The limited mathematical history we have from the East shows they might have been doing so much earlier.)

As early as 370 BC, integration techniques were used by mathematicians and scientists to solve problems with area and volume. However, most historical experts agree that the real groundwork was laid for integral calculus was laid by Bonaventura Cavalieri (pictured above) in his work with "indivisibles." He said that a line is just a sum of points, an area is just a sum of lines and an object is just a sum of areas. With this idea of dividing an area into an infinite number of sections, he was able to show that this method was effective in calculating the area of a triangle as well as the area under a parabola. (In fact, these divisions are very similar to the sums we now call "Riemann Sums.") He also was one of the first to utilize the idea of a limit, since he was estimating what would take place if the number of areas he used to calculate an area were to increase to infinity. Right after Cavalieri had been successful with his indivisibles, John Wallis came along and was able to do almost the same thing but with higher order polynomials. He was able to generalize a form for the area under a polynomial curve with vertex at (0,0). Pierre de Fermat was another mathematician who developed techniques that are part of the basis of integral calculus. He used infinite series to represent values such as - you guessed it - the area under a curve.

This mathematical basis was laid for the ability to solve problems involving what we know know as the integral, however, it wasn't until later that it was connected with the derivative and the fundamentals of calculus laid. The integral (under a variety of names) was used by scientists when circumstances required once the idea and methods for using these "infinitesimals" was formalized. It wasn't until Newton and Leibnitz found the connection between integrals and derivatives (that they undo each other), that integrals became the commonly used mathematical tool that it is today.