One of the uses of integration is to find the area between a curve and the xaxis.
These areas are found be evaluating a definite integral. When antidifferentiating the result is a function. When evaluating a definite integral the answer is a number.
Defiinite Integrals
A definite integral is calculated by integrating a function between two values, called the limits of integration. These two values are substituted into the integrated function and the difference taken.
The formula is:
f(x) dx = F(b) − F(a) where F(x) is an antiderivative of f(x) 
This complicated looking formula is easier to use than it looks. It simply means doing the following:
Method 
Example Evaluate (x^{2} + 3) dx 

Step 1  Integrate the function  = 
Step 2  Substitute the values 2 and 1 
=  
Step 3  Evaluate  
Note that the constant of integration c is not shown as it would be eliminated when subtracting. 
Definite Integration by Substitution
If the method of integration used is substitution, the limits of integration are changed to those of the variable used in the substitution.
Example