A series of mathematical techniques developed independently by Isaac Newton and Gottfried Leibniz (1646–1716). Differential calculus treats a continuously varying quantity as if it consisted of an infinitely large number of infinitely small changes. For example, the velocity v of a body at a particular instant can be regarded as the infinitesimal distance, written ds, that it travels in the vanishingly small time interval, dt; the instantaneous velocity v is then ds/dt, which is called the derivative of s with respect to t. If s is a known function of t, v at any instant can be calculated by the process of differentiation. The differential calculus is a powerful technique for solving many problems concerned with rate processes, maxima and minima, and similar problems.
Integral calculus is the opposite technique. For example, if the velocity of a body is a known function of time, the infinitesimal distance ds travelled in the brief instant dt is given by ds = vdt. The measurable distance s travelled between two instants t1 and t2 can then be found by a process of summation, called integration, i.e.s=∫t2t1vdtThe technique is used for finding areas under curves and volumes and other problems involving the summation of infinitesimals.