I recently learned about the formula for finding the surface area of a function rotated about the x-axis:
$$S = \int 2\pi y \sqrt{1+\bigg[ \frac{dy}{dx} \bigg]^2}dx$$
I tried to apply this formula to a problem involving the function $y = x^2$ rotated around the x-axis. The standard approach seems to involve taking a small frustum element and integrating over the limits for the derivation of the above formula.
However, when I first attempted (without knowing the formula) to set up the integral for this problem, I used the lateral surface area of a disc as the differential element, which resulted in the integral $\int 2\pi y dx$ rather than the one above. This approach was based on my previous experience with volume of revolution problems for the same function, where we used washers/discs to set up the integral, and the volume was given by $\int \pi y^2 dx$.
My question is: why is it acceptable to use a disc as the differential element when approximating a volume element, but not when calculating surface area (ie. rather than using a frustom as an area element why can't we use discs as done in a volume of revolution problem)? What is the fundamental difference between these two scenarios that necessitates a different approach?
