Calculus

Definition

To find the area under the curve you approximate with rectangles and as number of rectangles you get an exact result.

To approximate the area under on you can split into strips which will have width . This divides the interval into where , so , .

Let be any point in . Then . Provided the limit exists any choice of will yield the same result.

Using the Epsilon-Delta Definition of a Limit you get:


Properties

If is Continuous on or has a finite number of jump discontinuities then is integrable on


Symmetry

Suppose is continuous on then:

  1. is even
  2. is odd

Comparisons

  1. (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.

using

If is differentiable with range and is continuous on

Integration By Parts


Improper integrals

The integral is improper if:

  1. One or both of the limits are or
  2. is undefined at any point in

To find , consider and compute the limit (if it converges) as For an integral , split into 2 integrals: ,

Mean Value

The mean value of a function in the interval is:

This is because the integral is taking infinite samples. If has mean value over the interval then:

  1. has mean value
  2. 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 and the Method of Differences to compute tricky summations.