How was calculus developed?

Answer 1

Calculus is based on the idea of the infinitesimal. To facilitate analysis curves were considered to consist of infinitesimally small straight line segments and volumes were considered to consist of infinitesimally thin cross sections. Some of these ideas date back to the ancient Greeks but it was not until the seventeenth century that they were crystallized into a set of rules. A good book about this is The Continuous and the Infinitesimal by John Bell.

If you go on a trip travelling at continuously changing speed, then it's the case that for each infinitesimal interval of time (dt) the speed doesn't change and the infinitesimal distance travelled (dx) must be v.dt where v is the velocity at that instant. The total distance is then the infinite sum of all these bits, and is represented by the integral sign which you can think of as a smoothed out "sigma". Also the instantaneous velocity v is dx/dt (by dividing dx=v.dt by dt). By considering examples of constant speed and constant acceleration on a bit of graph paper, it dawned on people that an integral can be represented as an area, and velocity as a slope (or rate of change of area) and formulas from getting from one to another were soon developed. They called these rules "calculus"