Iterated Integrals: Examples

1. Evaluate:

Anwser:
 

First, find the integral according to y, then put in the bounds:

Now, using the function above, find the integral according to x, and put in the bounds:

 

2. Using Iterated Integrals, prove that the area of a rectangle is simple one side multiplied by the other.
 

Anwser:

Use the following integral and solve to prove:

Because each difference represents a side, this proves that the area of a rectangle is simple one side multiplied by the other.