One of the uses of integration is to find the area between a curve and the x-axis. 
These areas are found be evaluating a definite integral. When anti-differentiating 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:

Y12_Definite_Integration__01.gif f(x) dx = F(b) − F(a)

where F(x) is an anti-derivative of f(x)

This complicated looking formula is easier to use than it looks. It simply means doing the following:



Evaluate Y12_Definite_Integration__02.gif(x2 + 3) dx

Step 1 Integrate the function =Y12_Definite_Integration__03.gif
Step 2 Substitute the values 2 and 1

Y12_Definite_Integration__04.gif - Y12_Definite_Integration__05.gif

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.