Why does the integral for surface area require an expression for arc length? (Solid of Revo.)

I'm a lowly Calculus II student here and I noticed something interesting/confusing.

The integral formula for the volume of a solid of revolution (SoR) performed with the disk method has the form $$V=\int\pi f(x)^2 \mathrm{d}x,$$ and the same holds for the formula for the area of a circle, but this time $$f(x)$$ is the radius. This makes a lot of intuitive sense.

From this, it would also make sense that the formula for the surface area of a SoR is the integral of the length of the circumference of a circle but with $$f(x)$$ as the radius again. However, this is not the case. The expression for surface area of a SoR is the circumference arc length.

Why is this the case?

Surface area of a cone of slant generator length $$l$$ maximum radius $$r$$ and constant cone angle $$\sin^{-1}\dfrac{r}{l}$$ is
$$=\pi r l$$
When cone angle changes ( $$r$$ changes with $$l$$) you should have curved slant area $$\int 2 \pi r dl$$