I'm about halfway through The Archimedes Codex, and it's been interesting as the two co-authors take turns - one is more involved with the investigation and history of the physical book, while the other is more an expert on ancient Greek mathematics and Archimedes in particular. At any rate, the latter presents a sketch of one version of Archimedes' quadrature of the parabola. Now, usually when you talk about that, you get the method of exhaustion version, which is sensible in comparison. I could do that, if given a large sheet of paper. But this is a different proof from Archimedes' Method -- a text known only from the Archimedes Codex, and thus lost to humanity for centuries. This 'new' proof involves imagining weighing different chunks of parabolas and triangles, and finding that they balance on an imaginary lever if you make your construction correctly. Even if you had given me the huge hint that you needed to balance things in this way, there is no way I would ever have come up with this. As weird as the proof looks to my modern eyes, though, it's hard not to consider it an elegant weapon, for a more civilized age.
In contrast, calculus is the blaster. Sure, integrate 2x-x^2 from 0 to 2. gets you x^2 - x^3/3 or 4 - 8/3 is 4/3. Tada. The stormtrooper is dead, but you do not look cool and mystifying doing it. In contrast, Archimedes is the Yoda of geometry.