/sci/ - Science & Math

>>10307456
Calculus was born from attempts at solving two different and (seemingly) unrelated problems: first, the problem of calculating the areas and volumes of non-rectilinear figures (for example, a circle or a sphere); secondly, the problem of finding the tangent to any given curve. The first problem (finding areas) led to Integral Calculus, while the second (finding tangents) led to Differential Calculus.

Integral Calculus can be said to be older than Differential Calculus since it can be traced back to Archimedes' works which used the so-called "method of exhaustion", which was basically a very primitive method of calculating limits of sequences. The method of exhaustion itself is probably even older than Archimedes, since it is found in Euclid and was probably discovered by Eudoxus even earlier.

Differential Calculus is much younger since before the invention of Analytic Geometry the problem of finding tangents to curves wasn't really felt that much. Fermat and Descartes developed some rudimentary methods to find tangemts at certain classes of curves but it was with Leibniz and Newton that the ideas of using "infinitesimals" took root, and that idea was later developed into the modern concept of limits (chiefly by Cauchy). There exists also non-standard Analysis which rejects Cauchy's concept of limit in favor of the older idea of infinitesimals.

Even though Integral and Differential Calculus originated in order to solve two very specific geometric problems, it was soon realized that the methods of Calculus had much wider application. Integral Calculus can be used to describe how a certain variable quantity (which varies over time, or with respect to another independent variable) accumulates as the independent variable increases.

(to be continued)