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In calculus, the integral of a function is an extension of the concept of a sum. The process of finding integrals is called integration. The process is usually used to find a measure of totality such as area, volume, mass, displacement, etc., when its distribution or rate of change with respect to some other quantity (position, time, etc.) is specified. There are several distinct definitions of integration, with different technical underpinnings. They are, however, compatible; any two different ways of integrating a function will give the same result when they are both defined.
The integral of a real-valued function f of one real variable x on the interval [a, b] is equal to the signed area bounded by the lines x = a, x = b, the x-axis, and the curve defined by the graph of f. This is formalised by the simplest definition of the integral, the Riemann definition, which provides a method for calculating this area using the concept of limit by dividing the area into sucessively thinner rectangular strips and taking the sum of their areas.
Alternatively, if we let:
then the integral of f between a and b is a measure of S. In intuitive terms, integration associates a number with S that gives an idea about the 'size' of the set (but this is distinct from its Cardinality or order). This leads to the second, more powerful definition of the integral, the Lebesgue integral.
Leibniz introduced the standard long s notation for the integral. The integral of the previous paragraph would be written:
The sign represents integration, a and b are the endpoints of the interval, f(x) is the function we are integrating known as the integrand, and dx is a notation for the variable of integration. Historically, dx represented an infinitesimal quantity, and the long s stood for "sum". However, modern theories of integration are built from different foundations, and the notation should no longer be thought of as a sum except in the most informal sense. Now, the dx represents a differential form.
As an example, if f is the constant function f(x) = 3, then the integral of f between 0 and 10 is the area of the rectangle bounded by the lines x = 0, x = 10, y = 0, and y = 3. The area is the width of the rectangle times its height, so the value of the integral is 30. The same result can be found by integrating the function, though this is usually done for more complicated or smooth curves.
The most basic technique for computing integrals of one real variable is based on the fundamental theorem of calculus. It proceeds like this:
1. Choose a function f(x) and an interval [a, b].
2. Find an antiderivative of f, that is, a function F such that F' = f.
3. By the fundamental theorem of calculus, provided the integrand and integral have no singularities on the path of integration,
4. Therefore the value of the integral is F(b) - F(a).
Note that the integral is not actually the antiderivative, but the fundamental theorem allows us to use antiderivatives to evaluate definite integrals.
The difficult step is often finding an antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. More often, it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable.
Techniques include:
* Integration by substitution
* Integration by parts
* Integration by trigonometric substitution
* Integration by partial fractions
Even if these techniques fail, it may still be possible to evaluate a given integral. The next most common technique is residue calculus, whilst for nonelementary integrals Taylor series can sometimes be used to find the antiderivative. There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum.
Definite integrals may be approximated using several methods of numerical integration. One popular method, called the rectangle method, relies on dividing the region under the function into a series of rectangles and finding the sum. Other well-known methods are the trapezoidal rule and Simpson's rule.
Some integrals cannot be found exactly, and others are so complex that finding the exact answer would be extremely time-consuming or computationally intensive. Approximation, however, is a process which relies only on variable substitution, multiplication, addition, and division. It can be done easily and quickly by modern graphing calculators and computers. Many real-world applications of calculus rely on calculating integrals approximately because of the complexity of formulas and since an exact answer is unnecessary.
Several mathematical functions and constants can be defined by using an integral. The natural logarithm, denoted by ln(x), is defined for all x 0 as:
Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with or , which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.
The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675. He derived the integral symbol
from an elongated letter S, standing for summa (Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in 1822.
Niels Henrik Abel seems to have been the first to consider in a general way the question as to what differential expressions can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville. Cauchy early undertook the general theory of determining definite integrals, and the subject has been prominent during the 19th century. Frullani's theorem (1821), Bierens de Haan's work on the theory (1862) and his elaborate tables (1867), Dirichlet's lectures (1858) embodied in Meyer's treatise (1871), and numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke, Elliott, Leudesdorf, and Kronecker are among the noteworthy contributions.
Eulerian integrals were first studied by Euler and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows:
although these were not the exact forms of Euler's study.
If n is an integer, it follows that:
,
but the integral converges for all positive real n and defines an analytic continuation of the factorial function to all of the complex plane except for poles at zero and the negative integers. To it Legendre assigned the symbol G, and it is now called the gamma function. Besides being analytic over the positive reals, G also enjoys the uniquely defining property that logG is convex, which aesthetically justifies this analytic continuation of the factorial function over any other analytic continuation. To the subject Dirichlet has contributed an important theorem Liouville, 1839), which has been elaborated by Liouville, Catalan, Leslie Ellis, and others. On the evaluation of G(x) and logG(x) Raabe (1843-44), Bauer (1859), and Gudermann (1845) have written. Legendre's great table appeared in 1816.
And also, I am ALWAYS RIGHT.
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