The Relationship between the Surface Area of a Curve and A Circle's Circumference

In my BC Calc class, we are learning about the formula for finding the surface area of curves of revolution. I don't like just memorizing formulas without understanding why they work, so I have set out to discover how the formula for the surface area of a curve was derived.

I'm concerned as to why the formula does not represent the sum of the circumference of the infinite number of circles formed by the curve, rather than the sum of surface area of an inifinite number of cylinders. When I solve problems using my hypothetical method I get a different answer than I would using the formula I have been taught.

Can anyone please help me to understand what is wrong with the formula I made up?(Happy)