Integral

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An integral is a mathematical construction used in calculus to represent the area of a region in a plane bounded by the graph of a function in one real variable. Integrals use the following notation:

where a and b represent the lower and upper bounds of the interval being integrated over, f(x) represents the function being integrated (the integrand), and dx represents a dummy variable given various definitions, depending on the context of the integral.

There are two types of integrals. Definite integrals are integrals that are evaluated over limits of integration. Indefinite integrals are not evaluated over limits of integration. Evaluating an indefinite integral yields the antiderivative of the integrand plus a constant of integration.

Integration has many physical applications. The indefinite integral of an acceleration function with respect to time gives the velocity function defined to within a constant, while the definite integral of an acceleration function with respect to time gives the change in velocity between the upper and lower limits of integration. Likewise, the indefinite integral of a time function of velocity with respect to time gives the position function defined to within a constant, and the definite integral of this velocity function will give the change in position between the two limits of integration.

Integration is the inverse function of the derivative and the two notions are related by the Fundamental Theorem of Calculus.

The concept of integration can be extended to functions in more than one real variable, as well as functions defined over the complex numbers.

Properties of integrals

Intergation has the following properties[1]

  • ,

Anti-derivative

Most students struggle with the important difference between the anti-derivative and integration. An anti-derivative of a function is a function such that,

The integral of a function can be evaluated using its antiderivative,

This works for the kind of functions encountered in late high school and early university mathematics. It is, however, an incomplete method. For example one cannot write the anti-derivative of in terms of familiar functions (such as trigonometric functions, exponentials, and logarithms) and function operations.

Riemann integral

As a geometric interpretation of the integral of the area of a curve, the Riemann integral consists of dividing the area under the curve of the function into rectangles. The domain of the function is partioned into N segments of width . The height of the segment is dependent on which side of the rectangle is taken. The lower sum takes the lower side of the rectangle, the upper sum the higher side of the rectangle. In the limit of these two series become the integral. If they approach the same value then the integral exists, otherwise it is undefined.

Lebesgue Integral

The Lebesgue integral is usually introduced in late university or early postgraduate mathematics. It is naively described as rotating the Reimann integral, in that it is the range instead of the domain that is partitioned. An understanding of measure theory is required to understand this techniques.

See Also

External Links

Integrals - Wolfram MathWorld

References