Integration is often called antidifferentiation. It is equivalent to creating an infinite number of infinitely small rectangles that reach from the x-axis to the curve and approximating the sum of their areas. While you can find a proof of the rectangles with the "I'm Feeling Lucky Button" on google I'll quickly cover integrals.
There are two main types of integrals Indefinite and Definite:
Indefinite: These integrals are used to find the antiderivative of a function.
- As I'm sure you know, the derivative of x^2 is 2x. Therefore, the antiderivative of 2x is x^2 + C. Why the + C at the end? Because the derivative of x^2 plus any constant would be 2x as the derivative of a constant is zero. The big C represent a "constant".
Definite integrals: These integrals are used for finding a value for an area under a curve.
Following Part 1 of the Fundamental Theorem of Calculus :
The definite integral of f(x) from a lower limit of integration a to an upper limit of integration b is equivalent to F(b) - F(a) where F(x) is the antiderivative of f(x).
So how on earth is that useful? When finding the area under a curve from x = a to x = b all we need to do is find the definite integral of the function ( f(x) ) from lower limit a to upper limit b. I hope this can get you started.
This website should be able to help you as well. Integrals -- from Wolfram MathWorld
Have a good one. Good luck.