Area under curve is found by integration............rate of change is found by differentials.......they undo each other and are kind of opposites..........connected.....

Integral

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This article is about the concept of definite integrals in calculus. For the indefinite integral, see antiderivative. For the set of numbers, see integer. For other uses, see Integral (disambiguation).

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A definite integral of a function can be represented as the signed area of the region bounded by its graph.

Part of a series of articles about
Calculus
Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem
Differential[show]
Integral[hide] Lists of integrals Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integration by Parts Discs Cylindrical shells Substitution (trigonometric) Partial fractions Order Reduction formulae
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v t e

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral

${\displaystyle {\int }_a^{b}f(x)dx}$

is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.

Roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written:

${\displaystyle F(x)=\int f(x)dx.}$

The integrals discussed in this article are those termed definite integrals. It is the fundamental theorem of calculus that connects differentiation with the definite integral: if f is a continuous real-valued function defined on a closed interval[a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by

${\displaystyle {\int }_a^{b}f(x)dx=F(b)-F(a).}$

The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century, who thought of the integral as an infinite sum of rectangles of infinitesimal width. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space.

Contents

  [hide] 

• 1History
  • 1.1Pre-calculus integration
  • 1.2Newton and Leibniz
  • 1.3Formalization
  • 1.4Historical notation
• 2Applications
• 3Terminology and notation
  • 3.1Standard
  • 3.2Meaning of the symbol dx
  • 3.3Variants
• 4Interpretations of the integral
• 5Formal definitions
  • 5.1Riemann integral
  • 5.2Lebesgue integral
  • 5.3Other integrals
• 6Properties
  • 6.1Linearity
  • 6.2Inequalities
  • 6.3Conventions
• 7Fundamental theorem of calculus
  • 7.1Statements of theorems
    • 7.1.1Fundamental theorem of calculus
    • 7.1.2Second fundamental theorem of calculus
  • 7.2Calculating integrals
• 8Extensions
  • 8.1Improper integrals
  • 8.2Multiple integration
  • 8.3Line integrals
  • 8.4Surface integrals
  • 8.5Contour integrals
  • 8.6Integrals of differential forms
  • 8.7Summations
• 9Computation
  • 9.1Analytical
  • 9.2Symbolic
  • 9.3Numerical
  • 9.4Mechanical
  • 9.5Geometrical
• 10See also
• 11Notes
• 12References
• 13External links
  • 13.1Online books

History[edit]

See also: History of calculus

Pre-calculus integration[edit]

The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus (ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle.

A similar method was independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere (Shea 2007; Katz 2004, pp. 125–126).

The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his method of Indivisibles, and work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of xⁿ up to degree n = 9 in Cavalieri's quadrature formula. Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers.