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In calculus, the integral of a function is an extension of the concept of a sum, and is identified or "found" through the use of integration. The process is usually used to find a measure of totality in relation to another quantity by the comparison of a known measure such as area, volume or mass, and its distribution or rate of change with respect to some other continuous quantity such as time or a known position.
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.
For 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.
If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. If the domain of the function to be integrated is the set of real numbers, and if the region of integration is an interval, then the greatest lower bound of the interval is called the lower limit of integration, and the least upper bound is called the upper limit of integration.
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.
The term "integral" may also refer to antiderivatives. Though they are closely related through the fundamental theorem of calculus, the two notions are conceptually distinct. When one wants to clarify this distinction, an antiderivative is referred to as an indefinite integral (a function), while the integrals discussed in this article are termed definite integrals.
The first rigorous definition of the integral was created by Bernhard Riemann in 1854 and is known as the "Riemann integral". This definition provides a method for calculating the area under a curve using the concept of limit by dividing the area into successively thinner rectangular strips and taking the limiting value approached by the sum of their areas as the width of the strips approaches zero.
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. The Lebesgue integral was created by Henri Lebesgue to integrate a wider class of functions and to prove very strong theorems about interchanging limits and integrals (see Lebesgue's dominated convergence theorem).
Although the Riemann and Lebesgue integrals are the most important definitions of the integral, a number of others exist, including:
The most basic technique for computing integrals of one real variable is based on the fundamental theorem of calculus. It proceeds like this:
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:
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. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral.
Computations of volumes of solids of revolution can usually be done with disk integration or shell integration.
Specific results which have been worked out by various techniques are collected in the list of integrals.
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.
Main article: Symbolic integration
Many professionals, educators, and students now use computer algebra systems to make difficult (or simply tedious) algebra and calculus problems easier. The design of such a computer algebra system is nontrivial as systematic methods of antidifferentiation are difficult to formulate, although in many cases a definite integral can be computed without finding an antiderivative.
One difficulty in computing definite integrals is that it is not always possible to find "explicit formulae" for antiderivatives. For instance, there is a (nontrivial) proof that there is no elementary function (e.g., involving sin, cos, exp, polynomials, roots and so on) whose derivative is x<sup>x</sup>. As such, computerized algebra systems have no hope of being able to find an antiderivative for this particular function. Unfortunately, functions that have nice antiderivatives are the exception. If one writes a large random expression involving exponentials and polynomials, the odds are almost nil that it will have a "nice" antiderivative. (This statement can be made formal, but it is difficult to do so.)
One of the difficulties is to decide what set of functions to use as building blocks for antiderivatives. Usually, we need a set of antiderivatives closed under, say, multiplication and composition. This set of antiderivatives should also include polynomials, perhaps quotients, exponentials, logarithms, sines and cosines. The Risch-Norman algorithm is able to compute any integral of such a shape; that is, if the antiderivative involves polynomials, sines, cosines, etc..., the Risch-Norman algorithm will be able to compute it. Extended versions of this algorithm are implemented in Mathematica and the Maple computer algebra system.
Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the special functions of physics (like the Legendre functions, the hypergeometric function, the Gamma function and so on). Extending the Risch-Norman algorithm so that it includes these functions is possible but challenging.
Most humans are not able to integrate such general formulae, so in a sense computers are more skilled at integrating highly complicated formulae. On the other hand, very complex formulae are unlikely to have closed-form antiderivatives, so this advantage is dubious.
Not all integrals can be evaluated using a single limit process. An integral which can only be evaluated by considering it as the limit of integrals on successively larger and larger intervals is called an improper integral.
An improper integral is the limit of a definite integral, as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.
Improper integrals usually turn up when the range of the function to be integrated is infinite or, in the case of the Riemann integral, when the domain of the function is infinite. One common example of an improper integral is the Cauchy principal value.
Integrals can be taken over regions other than intervals. In general, an integral over a set E of a function f (where the domain of f includes E) is written:
Here x need not be a real number, but can be other suitable algebraic quantities. For instance, a vector in R<sup>3</sup>. Fubini's theorem shows that such integrals can be rewritten as an iterated integral. In other words, the integral can be calculated by integrating one coordinate at a time.
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane which contains its domain. (Note that the same volume can be obtained via the triple integral — the integral of a function in three variables — of the constant function f(x, y, z) = 1 over the above-mentioned region between the surface and the plane.) If the number of variables is higher, one will calculate "hypervolumes" (volumes of solid of more than three dimensions) that cannot be graphed. For example, the volume of the parallelepiped of sides 4×6×5 may be obtained in two ways:
Because it is impossible to calculate the antiderivative of a function of more than one variable, indefinite multiple integrals do not exist so they are all definite integrals.
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.[1] In Arabic which is written from right to left, an inverted integral symbol is used.
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