Integral

An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive. The Riemann integral is the simplest integral definition and the only one usually encountered in physics and elementary calculus. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."

The Riemann integral of the function  over  from  to  is written

(1)

Note that if , the integral is written simply

(2)

as opposed to .

Every definition of an integral is based on a particular measure. For instance, the Riemann integral is based on Jordan measure, and the Lebesgue integral is based on Lebesgue measure. Moreover, depending on the context, any of a variety of other integral notations may be used. For example, the Lebesgue integral of an integrable function  over a set  which is measurable with respect to a measure  is often written

(3)

In the event that the set  in () is an interval , the "subscript-superscript" notation from (2) is usually adopted. Another generalization of the Riemann integral is the Stieltjes integral, where the integrand function  defined on a closed interval  can be integrated against a real-valued bounded function  defined on , the result of which has the form

(4)

or equivalently

(5)

Yet another scenario in which the notation may change comes about in the study of differential geometry, throughout which the integrand  is considered a more general differential k-form  and can be integrated on a set  using either of the equivalent notations

(6)

where  is the above-mentioned Lebesgue measure. Worth noting is that the notation on the left-hand side of equation () is similar to that in expression () above.

The process of computing an integral is called integration (a more archaic term for integration is quadrature), and the approximate computation of an integral is termed numerical integration.

There are two classes of (Riemann) integrals: definite integrals such as (5), which have upper and lower limits, and indefinite integrals, such as

(7)

which are written without limits. The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if  is the indefinite integral for , then

(8)

What's more, the first fundamental theorem of calculus can be rewritten more generally in terms of differential forms (as in () above) to say that the integral of a differential form  over the boundary  of some orientable manifold  is equal to the exterior derivative  of  over the interior of , i.e.

(9)

Written in this form, the first fundamental theorem of calculus is known as Stokes' Theorem.

Since the derivative of a constant is zero, indefinite integrals are defined only up to an arbitrary constant of integration , i.e.,

(10)

Wolfram Research maintains a web site http://integrals.wolfram.com/ that can find the indefinite integral of many common (and not so common) functions.

Differentiating integrals leads to some useful and powerful identities. For instance, if  is continuous, then

(11)

which is the first fundamental theorem of calculus. Other derivative-integral identities include

(12)

the Leibniz integral rule

(13)

(Kaplan 1992, p. 275), its generalization

(14)

(Kaplan 1992, p. 258), and

(15)

as can be seen by applying (14) on the left side of (15) and using partial integration.

Other integral identities include

(16)

(17)

			(18)
			(19)

and the amusing integral identity

(20)

where  is any function and

(21)

as long as  and  is real (Glasser 1983).

Integrals with rational exponents can often be solved by making the substitution , where  is the least common multiple of the denominator of the exponents.

REFERENCES:

Beyer, W. H. "Integrals." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 233-296, 1987.

Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, 2004.

Bronstein, M. Symbolic Integration I: Transcendental Functions. New York: Springer-Verlag, 1996.

Dubuque, W. G. "Re: Integrals done free on the Web." math-fun@cs.arizona.edu posting, Sept. 24, 1996.

Glasser, M. L. "A Remarkable Property of Definite Integrals." Math. Comput. 40, 561-563, 1983.

Gordon, R. A. The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Providence, RI: Amer. Math. Soc., 1994.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.

Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 29, 1988.

Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1992.

Piessens, R.; de Doncker, E.; Uberhuber, C. W.; and Kahaner, D. K. QUADPACK: A Subroutine Package for Automatic Integration. New York: Springer-Verlag, 1983.

Ritt, J. F. Integration in Finite Terms: Liouville's Theory of Elementary Methods. New York: Columbia University Press, p. 37, 1948.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 145, 1993.

Wolfram Research. "The Integrator." http://integrals.wolfram.com/.