It's not infinitesimal "reasoning" because infinitesimal "reasoning" is not reasoning. And it's EXACTLY why I say Archimedes understood calculus better [in a sense] than Newton did. He is not waving his arms frantically in the air about "reaching a point where the small quantity o is zero." He is actually showing you how to do integration, and in the course of doing the construction, he is asserting that
The intersection of an infinite number of nested non-empty closed subsets of a compact space is not empty.
That is the fundamental basis of Calculus, and does not involve the "reasoning" of "infinitesimals."
Archimedes constructions depend on finding exact expressions for the limiting values. This makes them idiosyncratic, because, supposing you can't?
The calculus derivation for the surface of a unit sphere is most definitely based on infinitesimal reasoning. The notation, d theta, in the integral is an expression of this. We have Int (d theta 2 pi sin theta) from 0 to pi, where d theta is the infinitesimal width of each strip of the surface in the integration, and 2 pi sin theta is the length of each circular strip. Of course this evaluates to 4 pi.
You have to use infinitesimal reasoning just to write this down!