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: 
Here I'd not argue with lots of formulas.  

Rather I'd provide the picture proof from Wikipedia:  

cf. <https://en.wikipedia.org/wiki/Area_of_a_circle#/media/File:CircleArea.svg>

--- rk
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 $0.5\times r\times2\pi r=\pi r^2$