When we calculate volumes of revolution, we typically approximate a small (infinitesimal?) distance $\mathrm{d}x$ or $\mathrm{d}y$, either along the $x$-axis or the $y$-axis. This partitioning does not take into account that the curve bends, and yet it yields the correct result in the end anyway.
However, when calculating the surface area of such a shape, it no longer suffices to use such a crude approximation, instead you need to use the length element $\mathrm{d}s$ which takes into account both the change in $x$ and $y$, as opposed to only one of them.
How come for volumes, we can safely ignore that the curve bends, whereas for surface areas, you are required to take that into account?
