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:

<math>\int_a^b f(x)dx</math>

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.

Properties of intergrals

Intergation has the following properties,

• Linerality <math>\int_{a}^{b}(f(x)+g(x))dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}g(x)dx</math>

• <math>\int_{a}^{b}cf(x)dx=c\int_{a}^{b}f(x)dx</math>

• <math>\int_{a}^{a}f(x)dx=0</math>

• <math>\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx</math>

• <math>\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx</math>

Anti-derivative

Most students struggle with the important difference between the anti-derivative and integration. The anti-derivative of a function <math>f(x)</math> is the function <math>g(x)</math> such that,

<math>\frac{d}{dx}g(x)=f(x)</math>

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

<math>\int_a^b f(x)dx=g(b)-g(a)</math>

<math>\int^\infty_{-\infty} f(x)=g(x)+C</math>

In the second case C is the constant of integration. As this is very common <math>\infty</math> and <math>-\infty</math> 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 <math>e^{x^{2}}</math> has no anti-derivative.

Riemann intergral

As a geometric interpretation of the integral of the area of a curve the Riemann integral consist of dividing the area under the curve of the function into rectangles.

See Also

• Methods of integration

External Links

Integrals - Wolfram MathWorld