I don't know that limit is the key factor here, but I agree on the calculus part. The arguments that rely on Cavalieri's principle to relate the volume of the

I don't know that limit is the key factor here, but I agree on the calculus part. The arguments that rely on Cavalieri's principle to relate the volume of the cone to the volume of a slice-by-slice identical pyramid, for example, don't really involve a limit per se, but they are clearly motivational for integral calculus. Basically every intuitional argument for this, and other volume formulas, are somewhere on the road to integral calculus.