Question: If you take the circumference of the circle and stretch it into a rope and then multiply it by the height(radius) why don't you get the area of a circle?

I ask this question when using the shell method revolved around the $x$ axis on $x^2$ from $[0,2] $ instead of the disk method.

Answer: 
Because if you join the ends of a small piece of your rope to the centre, you get a triangle, not a rectangle. A triangle has half the area of a rectangle with the same base and height.
You are stretching both the circumference and the center to make a rectangle of height$R$

By turning the circumference of the circle into a rope and keeping the center fixed you are going a triangle not a rectangle so the area is $$ (1/2)(2\pi R)(R) = \pi R^2 $$