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: 
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 $$ (1/2)(2\pi R)(R) = \pi R^2 $$
In your logic, you are saying that unraveling a circle gives you many small rectangles. How does that make any sense when you try to roll it back? 

Instead, it comes out as many triangles of height $r$, and sum of all the base lengths would be $2\pi r$(the circumference). Hence their total area is $0.5\times r\times2\pi r=\pi r^2$