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For other meanings of "integral", see integration.
It is recommended that the reader be familiar with algebra, derivatives, functions, and limits.
In mathematics, the term "integral" has two unrelated
meanings; one relating to integers, the other relating to integral calculus.
"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):a≤x≤b,0≤y≤f(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:
- Choose a function f(x) and an interval [a,b].
- Find an antiderivative of f, that is, a function
F such that F' =f.
- By the Fundamental Theorem of Calculus, .
- 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 use of 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.
Wikipedia provides a list of integrals for the curious,
stumped, or lazy. Don't do your homework with them. You'll fail the test.
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:
- the Daniell
integral
- the Darboux integral,
a variation of the Riemann integral
- the Denjoy integral, an
extension of both the Riemann and Lebesgue integrals
- the Haar integral
- the Henstock-Kurzweil integral, an
extension 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 (also
called Lebesgue-Radon integral)
- the Perron integral,
which is equivalent to the restricted Denjoy integral
- the Riemann-Stieltjes integral, an
extension of the Riemann integral
See also
External links
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