Defintion of Integration - Indefinite Integral

 

A function F(x) is called antiderivation of f(x) when it is:

F(x) is also called integral of f(x). The procedure of determining the integral is called integration.

Since the derivation of a constant C is zero, then for a function there are an infinite number of integrals:

The aggregate of the antiderivatives of a function is called indefinite integral. The notation of the indefinite integral is:

Integration and derivation are opposite procedures. So,

 

Geometric Interpretation-Definite Integral

Consider the following function y=f(x). In order to calculate the area S between the points a and b on the X axis and the curve, we divide the length b-a on the X axis in n parts of length Δx. Then we can multiply Δx by the value of f(x) in the middle m of Δx to find approximately the area of each parallelogram. Decreasing the magnitude of Δx, the approximation becomes more accurate. Summing the area of all parallelograms provides us the total area.

Image 1:Definite Integral

 

Assuming that n reaches infinity, then Δx reaches zero which eliminates the approximation error. In this case the above equation becomes:

So integration of a function is the calculation of the area between this function and the X axis. When this calculation applies between two points, the integration is called definite integration. The indefinite integration returns a function and the definite integration returns a number.