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Integral
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Integral

For other meanings of "integral", see integration.
It is recommended that the reader be familiar with algebra, derivatives, functionss, and limitss.

Topics in Calculus
Fundamental theorem | Function | Limits of functions | Continuity | Calculus with polynomials
Differentiation
Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem
Integration
Integration by substitution | Integration by parts | Integration by trigonometric substitution | Solids of revolution | Integration by disks | Integration by cylindrical shells | Lists of integrals
Vector Calculus
Vector | Vector field | Matrix | Partial Derivative | Gradient | Flux | Divergence | Divergence Theorem | Del | Curl | Stokes' Theorem

In mathematics, the term "integral" has two distinct meanings; one relating to integers, the other relating to '\integral calculus'.

Table of contents
1 "Integral" in relation to integers
2 Integral calculus
3 Computing integrals
4 Improper integrals
5 Definitions of the integral
6 See also
7 External links

"Integral" in relation to integers

A real number is "integral" if it is an integer.

The integral value of a real number x is defined as the largest integer which is less than, or equal to, x. The integral value of x is often denoted by ; and called the "floor function".

In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. Integral domains are generalizations of the integers.

In number theory, an element of a number field is called integral if it is an algebraic integer.

Integral calculus

In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. Unlike the process of differentiation, there are several different definitions of integration, all of which have different technical underpinnings. However, any two different ways of integrating a function will give the same result if they are both defined.

Intuitively, the integral of a continuous, positive real-valued function f of one real variable x between a left endpoint a and a right endpoint b represents the area bounded by the lines x=a, x=b, the x-axis, and the curve defined by the graph of f. More formally, if we let S={(x,y):axb,0≤yf(x)}, then the integral of f between a and b is the measure of S.

Leibniz introduced the standard long s notation for the integral. The integral of the previous paragraph would be written . The ∫ sign represents integration, the a and b are the endpoints of the interval, f(x) is the function we are integrating, and dx is 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 traditional symbols have become no more than notation.

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 10c, so the value of the integral is 30.

Integrals can be taken over regions other than intervals. In general, the integral over a set E of a function f is written ∫Ef(x)dx. Here x need not be a real number, but, for instance, a vector in R3. 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.

If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. Integrals are sometimes called definite integrals to emphasize that they result in a number, not another function. This is to distinguish them from indefinite integrals, which are another name for an antiderivative. If the domain of the function is the 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.

Computing integrals

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, .
  4. Therefore the value of the integral is F(b)-F(a).

Note that the integral is not actually the antiderivative (it is a number), but the fundamental theorem allows us to use antiderivatives to evaluate integrals.

The difficult step is 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 the integral. The next most common technique is residue calculus. There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform the integral of a square into an infinite sum. Occasionally an integral can be evaluated by a trick; for an example of this, see Gaussian integral.

Computation 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.

Approximation of definite integrals

Definite integrals may be approximated using several methods. One popular method, called the rectangle method or the trapezoidal rule, relies on dividing the function into a series of rectangles and finding the sum. Another well-known method is 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 integral approximation because of the complexity of formulas and unnecessary nature of an exact answer.

Integrals and computerized algebra systems

Many professionals, educators, and students now use computerized 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.

One difficulty is that it is not always possible to find "nice formulae" for antiderivatives. For instance, there is a (nontrivial) proof that there is no nice function (e.g., involving sin, cos, exp, polynomials, roots and so on) whose derivative is exp(-x2). 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 an 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 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.

Improper integrals

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 integrals is called an improper integral. Improper integrals usually turn up when the range of the function is infinite or, in the case of the Riemann integral, when the domain is infinite. One common example of an improper integral is the Cauchy principal value.

Definitions of the integral

The most important integrals are the Riemann integral and the Lebesgue integral. The Riemann integral was created by Bernhard Riemann and was the first rigorous definition of the 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.

Although the Riemann and Lebesgue integrals are the most important ones, a number of others exist, including but not limited to:

See also

External links