Integral

This article deals with the concept of an integral in mathematical calculus . For other meanings of "integral" see integration .

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 whichhave different technical underpinnings. However, any two different ways of integrating a function will give the same result ifthey are both defined.

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

Leibniz introduced the standard long s notation for the integral. The integral of the previous paragraph would be written . The ∫ sign represents integration, the aand b are the endpoints of the interval , f(x) is the function weare integrating, and dx is notation for the variable of integration. Historically, dx represented aninfinitesimal quantity, and the long s stood for "sum". However, modern theories of integration are built from differentfoundations, 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 thearea 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 functionf 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, theintegral 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 iscalled the integrand. Integrals are sometimes called definite integrals to emphasize that theyresult in a number, not another function. This is to distinguish them from indefinite integrals, which areanother name for an antiderivative . If the domain of the function isthe 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 proceedslike this:

1. Choose a function f(x) and an interval [a,b].
2. Find an antiderivative of f, that is, a functionF 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 useantiderivatives to evaluate integrals.

The difficult step is finding an antiderivative of f. It is rarely possible to glance at a function and write downits 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 trigonometricsubstitution
• Integration by partialfractions

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 commonways 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 usuallybe 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 ofrectangles and finding the sum. Another well-known method is Simpson'srule .

Some integrals cannot be found exactly, and others are so complex that finding the exact answer would be extremelytime-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. Manyreal-world applications of calculus rely on integral approximation because of the complexity of formulas and unnecessary natureof 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 calculusproblems easier. The design of such a computer algebra system is nontrivial as systematic methods of antidifferentiation aredifficult 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 derivativeis exp(-x²). As such, computerized algebra systems have no hope of being able to find an antiderivative forthis particular function. Unfortunately, functions that have nice antiderivatives are the exception. If one writes a large randomexpression involving exponentials and polynomials, the odds are almost nil that it will have an antiderivative. (This statementcan 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 aset of antiderivatives closed under, say, multiplication and composition. This set of antiderivatives should also includepolynomials, perhaps quotients, exponentials, logarithms, sines and cosines. The Risch-Norman algorithm is able to compute any integral of such a shape; that is, if theantiderivative involves polynomials, sines, cosines, etc..., the Risch-Norman algorithm will be able to compute it. Extendedversions of this algorithm are implemented in Mathematica and 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 ofantiderivatives, the special functions of physics (like the Legendrefunctions , the Hypergeometric function , the Gamma function and so on.) Extending the Risch-Norman algorithm so thatit 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 highlycomplicated formulae. On the other hand, very complex formulae are unlikely to have closed-form antiderivatives, so thisadvantage 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 asthe 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 limitedto:

• the Daniellintegral
• the Darboux integral , a variation of the Riemann integral
• the Denjoy integral , anextension of both the Riemann and Lebesgue integrals
• the Haar integral
• the Henstock-Kurzweil integral , anextension of both the Riemann and Lebesgue integrals (also called HK-integral)
• the Henstock-Kurzweil-Stieltjes integral (also called HK-Stieltjes integral)
• the Lebesgue-Stieltjes integral (alsocalled Lebesgue-Radon integral)
• the Perron integral ,which is equivalent to the restricted Denjoy integral
• the Riemann-Stieltjes integral , anextension of the Riemann integral

See also

• calculus
• list of integrals
• derivatives
• limits .
• functions
• algebra

External links

• The Integrator by Wolfram Research
• Function Calculator from WIMS