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Differential calculus and integral calculus are connected by the fundamental theorem of calculus. This states that differentiation is the reverse process to integration . Differentiation has applications in nearly all quantitative disciplines.
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus.
Fundamental theorem of calculus; Integration by parts; Inverse chain rule method; Integration by substitution. Tangent half-angle substitution; Differentiation under the integral sign; Trigonometric substitution; Partial fractions in integration. Quadratic integral; Proof that 22/7 exceeds π; Trapezium rule; Integral of the secant function ...
From the conjecture and the proof of the fundamental theorem of calculus, calculus as a unified theory of integration and differentiation is started. The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, [ 2 ] was by James Gregory (1638–1675).
Integration, the process of computing an integral, is one of the two fundamental operations of calculus, [a] the other being differentiation. Integration was initially used to solve problems in mathematics and physics , such as finding the area under a curve , or determining displacement from velocity.
In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).
The skew-symmetry of differential forms means that the integral of, say, dx 1 ∧ dx 2 must be the negative of the integral of dx 2 ∧ dx 1. Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined.
Meanwhile, calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the study of continuous change. Discrete calculus has two entry points, differential calculus and integral calculus. Differential calculus concerns incremental rates of change and the slopes of piece-wise linear curves.