The previous topic stated that integration is the opposite of differentiation. Hence its other name of antidifferentiation.
While this is true, one of the main uses of integration is to find the area between a curve and the xaxis.
These areas are found by evaluating a definite integral.
Defiinite Integrals
A definite integral is calculated by integrating a function between two values. 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 the 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. 
Area under a Curve
An accurate way to find the area between a curve, two xvalues and the xaxis, as shaded in the example on the diagram below, is to find the definite integral between those xvalues.
To find the shaded area between the curve f(x) = x^{2}6x + 10 , the lines x = 2 and x = 5 and the xaxis, evaluate the definite integral: (x^{2} − 6x + 10) dx
The shaded area is 6 square units.

It is advisable although not essential to draw a diagram before carrying out the integration.
Sometimes, when the curve crosses the xaxis, part of the area may be above the xaxis and part below the xaxis.
When this happens evaluate each area separately, treating both as positive, and then add them together.