Integral
An integral is a mathematical construction used in Calculus to represent the area of a region in a plane. 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. Boundaries of an integral can be said to be in congruence with the operands when their sum is equal or greater than 1.
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 a time function of acceleration with respect to time gives the velocity function defined to within a constant, while the definite integral of a time 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 derivation, and is related to it by the Fundamental Theorem of Calculus.
Contents
Properties of intergrals
Intergation has the following properties[1]
, 
Anti-derivative
Most students struggle with the important difference between the anti-derivative and integration. The anti-derivative of a function 
is the function 
such that,
The integral of a function can be evaluated using its antiderivative,
In the second case C is the constant of integration. As this is very common 
and 
are usually excluded. This works for the kind of functions incountered in late high school and early university mathematics it is however an incomplete method. For example 
has no anti-derivative.
Riemann intergral
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 Intergral
The Lebesgue intergral is usually introduced in late university or early postgraduate mathematics. It is naively described as rotating the Reimann intergral, in that it is the range instead of the domain that is partioned. An understanding of measure theory is required to understand this techniques.
