$\begingroup$

Before I read the formula of the area of revolution which is $\int 2\pi y \,ds$, where $ds = \sqrt{1 + \frac{dy}{dx}^2}$, I thought of deriving it myself. I tried to apply the same logic used for calculating the volume of revolution (e.g., $\int \pi y^2 dx $).

My idea is to use many tiny hollow cylinders (inspired from the shell method), each has a surface area of $(2\pi y) (dx)$:

  • $2\pi y$ is the circumference of the cylinder, and
  • $dx$ is the height of the cylinder

Their product is the surface area of the hollow (e.g., empty from the inside) cylinder.

With this logic, the area is $\int 2\pi y dx$.

Where is my mistake? Also it's confusing why for the volume it was enough to partition the object using cylinders and for areas not.