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.
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 anti-derivative of f(x)
This complicated looking formula is easier to use than it looks. It simply means doing the following:
Evaluate (x2 + 3) dx
|Step 1||Integrate the function||=|
|Step 2||Substitute the values 2 and 1||
|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.