[sjmiller]

"Known as the founder of integral calculus and mathematical physics". I'm a mathematician, and it's a bit of a stretch to call him the founder of integral calculus; that's usually reserved for Newton and Leibnitz. Archimedes came extremely close, as evidenced by his approximation of pi by looking at inscribed and circumscribed polygons about the circle.

[dr_lew]

Archimedes' derivation of the formula for the volume of a cone was essentially integration. The factor of 1/3 is from the integral of pi r squared dr.

He used infinitesimal reasoning in a recognizable fashion, but always in an ad hoc geometrical argument, so he had a technique, but not a generalized method.