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.
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