5. Calculus Calculus is another vast discipline associated with the study of conditions of continuous change and its impact on functions. It has two major branches:
Differential calculus deals with the rate of change and slope of continuous functions. For example, biologists determine the rate of growth in bacterial culture with the help of differential calculus.
Integral calculus involves calculating accumulation of entities and areas under curves. It is often used by architects to calculate the quantity of materials required for curved surfaces.
Text Integral and Differential calculus Calculus is a branch of mathematics using the idea of a limit and generally divided into two parts: integral and differential calculus.
Integral and differential calculus can be used for finding areas, volumes, lengths of curves, centroids, and moments of inertia of curved figures. It can be traced back to Eudoxus of Cnidus and his method of exhaustion (c. 360 ВС). Archimedes (in “The Method”) developed a way of finding the areas of curves by considering them to be divided up by many parallel line segments, and extended it to determine the volumes of certain solids; for this he is sometimes called the “father of the integral calculus”.
In the early 17th century interest again developed in measuring volumes by integration methods. Kepler used a procedure for finding the volumes of solids by taking them to be composed of an infinite set of infinitesimally small elements (“Measurement of the Volume of Barrels”, 1615). These ideas were generalized by Cavalieri in his “Geometria indivisibilibus continuorum nova” (1635), in which he used the idea that an area is made up of indivisible lines and a volume of indivisible areas; i.e., the concept used by Archimedes in “The Method”. Cavalieri thus developed what became known as his “method of indivisible”. John Wallis, in “Arithmetica infinitorum” (1655) arithmetized Cavalieri's ideas. In this period infinitesimal methods were extensively used to find lengths and areas of curves.
Differential calculus is concerned with the rates of changes of functions with respect to changes in the independent variable. It came out of problems of finding tangents to curves, and an account of the method is published in Isaac Barrow's “Lectiones geometricae” (1670). Newton had discovered the method (1665-66) and suggested that Barrow include it in his book. In his original theory Newton regarded a function as a changing quantity - a fluent - and the derivative or rate of change he called a fluxion. The slope of a curve at a point was found by taking a small element at the point and finding the gradient of a straight line through this element. The binomial theorem was used to find the limiting case,
e., Newton's calculus was an application of infinite series. He used the notation x' and y' for fluxions and x” and y” for fluxions of fluxions. Thus, if x= f(t), where x is the distance and t - the time for a moving body, then x' is the instantaneous velocity and x” - the instantaneous acceleration. Leibniz had also discovered the method by 1676 publishing it in 1684. Newton did not publish until 1687. A bitter dispute arose over the priority for the discovery. In fact it is now known that the two made their discoveries independently and that Newton made his about ten years before Leibniz, although Leibniz published first. The modern notation of dy/dx and the elongated s for integration is due to Leibniz.
From about this time integration came to be regarded simply as the inverse process of differentiation. In the 1820s Cauchy put the differential and integral calculus on a more secure footing by using the concept of a limit. Differentiation he defined by the limit of a ratio and the integration by the limit of a type of sum. The limit definition of integral was made more general by Riemann.
In the 20th century the idea of an integral has been extended. Originally integration was concerned with elementary ideas of measure (i.e., lengths, areas and volumes) and with continuous functions. With the advent of set theory functions came to be regarded as one-to-one mappings, not necessarily continuos, and more general and abstract concepts of measure were introduced. Lebesque put forward a definition based on the Lebesque measure of a set. Similar extensions of the concept have been made by other mathematicians.
