Integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one
of the two main operations in calculus. Given a function f of a real variable x and an interval [a, b] of
the real line, the definite integral
is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis,
and the vertical lines x = a and x = b, such that areas above the axis add to the total, and the area below
the x axis subtract from the total.
The term integral may also refer to the notion of antiderivative, a function F whose derivative is the
given function f. In this case, it is called an indefinite integral and is written:
The integrals discussed in this article are termed definite integrals.The principles of integration were
formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Through the
fundamental theorem of calculus, which they independently developed, integration is connected with
differentiation: 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
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Integrals and derivatives became the basic tools of calculus, with numerous applications in science and
engineering. The founders of the calculus 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 which 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.
Integrals of differential forms play a fundamental role in modern differential geometry. These
generalizations of integrals first arose from the needs of physics, and they play an important role in the
formulation of many physical laws, notably those of electrodynamics.
There are many modern concepts of integration, among these, the most common is based on the
abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.
Pre-calculus integration :- 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 shapes 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. Similar methods were
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).
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