Definition
To find the area under the curve you approximate with rectangles and as number of rectangles
To approximate the area under
Let
Using the Epsilon-Delta Definition of a Limit you get:
Properties
If
Symmetry
Suppose
is even is odd
Comparisons
(integral is in between rectangles bounded by maximum and minimum)
Common Results
Techniques
Reverse Chain Rule
Substitution
Sometimes you can simplify an integral by changing the variable. This process is similar to using the chain rule in differentiation and is called integration by substitution.
If
Integration By Parts
Improper integrals
The integral
- One or both of the limits are
or is undefined at any point in
To find
Mean Value
The mean value of a function
This is because the integral is taking infinite samples.
If
has mean value has mean value ,
Reduction
The Reduction formula allows you to write an integral as a recurrence relation. This is generally used for integrals with high powers that would require many integration by parts iterations.
You can use the reduction formula in conjunction with a substitution