[stephenjohnbanker]

Calculus and Analysis Calculus Integrals
MathWorld Contributors Cortzen


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. 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 (ž), which have upper and lower limits, and indefinite integrals, such as

(3)

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

(4)

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

(5)

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

(6)

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

(7)

the Leibniz integral rule

(8)

(Kaplan 1992, p. 275), its generalization

(9)

(Kaplan 1992, p. 258), and

(10)

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

Other integral identities include

(11)

(12)

(13)
(14)

and the amusing integral identity



SO THERE!

[Irish_Thatcherite]

A square wave is the derivative of a triangular wave, and a triangular wave is the integral of a square wave, so an electronic circuit called an integrator converts square waves into triangular waves, and then of course there is Fourier and Laplace..... (shudder).

[NicknamedBob]

The Leibniz integral rule gives a formula for differentiation of a definite integral whose limits are functions of the differential variable,

It is sometimes known as differentiation under the integral sign.

This rule can be used to evaluate certain unusual definite integrals such as

for (Woods 1926).

Feynman (1997, pp. 69-72) recalled seeing the method in Woods (1926) and remarked "So because I was self-taught using that book, I had peculiar methods for doing integrals," and "I used that one damn tool again and again."

[NicknamedBob]

You were doing the equivalent of posting a reference which had been “earlier quoted” using the equivalent of “ibid.”

As you discovered, that doesn’t work.

What you have to do, in addition to copying all of the text and referred diagrams, is copy the original root or prefix to the referred diagrams and insert it into the image source reference.

Simple as pi, isn’t it?