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INTEGRAL CALCULUS

This article deals with the concept of an integral in calculus. For other meanings of "integral" see integration and integral (disambiguation).
Topics in calculus

Fundamental theorem
Limits of functions
Continuity
Vector calculus
Tensor calculus
Mean value theorem
Differentiation

Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Related rates
Table of derivatives
Integration

Lists of integrals
Improper integrals
Integration by: parts, disks,
cylindrical shells, substitution,
trigonometric substitution

In calculus, the integral of a function is an extension of the concept of a sum. The process of finding integrals is integration, in its mathematical meaning. The process is usually used to find a measure of totality such as area, volume, mass, displacement etc when its variation (height, density, velocity etc) with respect to some other quantity (position, time etc) is specified for values of that quantity. There are several possible definitions of integration, with different technical underpinnings. They are, however, compatible; any two different ways of integrating a function will give the same result when they are both defined.

The term "integral" may also refer to antiderivatives. Though they are closely related through the fundamental theorem of calculus, the two notions are conceptually distinct. When one wants to clarify this distinction, an antiderivative is referred to as an indefinite integral (a function), while the integrals discussed in this article are termed definite integrals.

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 is equal to the area bounded by the lines x = a, x = b, the x-axis, and the curve defined by the graph of f. This is formalised by the simplest definition of the integral, the Riemann definition, which provides an algebraic way of calculating this area using the concept of limit. If we let

S= \{(x,y) \in \mathbb{R}^2:a \leq x \leq b ,0 \leq y \leq f(x)\},

then the integral of f between a and b is the measure of S. In intuitive terms, it associates a number with S that gives an idea about the 'size' of the set (but this is distinct from its Cardinality or order). This leads to the second, more powerful definition of the integral, the Lebesgue integral.

Leibniz introduced the standard long s notation for the integral. The integral of the previous paragraph would be written \int_a^b f(x)\,dx. The ∫ sign represents integration, a and b are the endpoints of the interval, f(x) is the function we are integrating known as the integrand, and dx is a 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 notation should no longer be thought of as a sum except in the most informal sense. Now, the dx represents a differential form.

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 the width of the rectangle times its height, so the value of the integral is 30. The same result can be obtained formally by integrating the function over this interval. In fact, the well-known formula for the area of a rectangle can be derived using integration.


Contents

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, provided the integrand and integral have no singularities on the path of integration, \int_a^b f(x)\,dx = F(b)-F(a).
  4. Therefore the value of the integral is F(b) − F(a).

Note that the integral is not actually the antiderivative, but the fundamental theorem allows us to use antiderivatives to evaluate definite 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 a given integral. The next most common technique is residue calculus, whilst for nonelementary integrals Taylor series can sometimes be used to find the antiderivative. There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral.

Computations 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 of numerical integration. One popular method, called the rectangle method, relies on dividing the region under the function into a series of rectangles and finding the sum. Other well-known methods are the trapezoidal rule and 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 calculating integrals approximately because of the complexity of formulas and since an exact answer is unnecessary.

Symbolic integration

Main article: Symbolic integration

Many professionals, educators, and students now use computer 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, although in many cases a definite integral can be computed without finding an antiderivative.

One difficulty in computing definite integrals is that it is not always possible to find "explicit formulae" for antiderivatives. For instance, there is a (nontrivial) proof that there is no elementary function (e.g., involving sin, cos, exp, polynomials, roots and so on) whose derivative is xx. As such, computerized algebra systems have no hope of being able to find an antiderivative for this particular function unless. 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 a "nice" 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 Mathematica and 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 intervals is called an improper integral. Improper integrals usually turn up when the range of the function to be integrated is infinite or, in the case of the Riemann integral, when the domain of the function 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 in 1854 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 (see Lebesgue's dominated convergence theorem).

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

Definitions by means of an integral

Several mathematical functions and constants can be defined by using an integral. The natural logarithm is usually defined as the positive number x such that

\ln x = \int_1^x \! {dt\over t}.

The mathematical constant e may then be defined as the unique number satisfying

\ln e =\int_1^e \! {dt\over t} = 1.

Both the beta function and gamma function are defined in terms of integrals.

History of integral notation

Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with \dot{x} or x'\,\!, which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.

The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675. He derived the integral symbol, "∫", from an elongated letter S, standing for summa (Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in 1822.[1]

See also

References

  1. ^ Earliest Uses of Symbols of Calculus

External links

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