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Everyone that has taken advanced Calculus remembers those nightmarish problems of revolving curves around axes, and integrated the resulting volumes. An unlikely article in the New York Times discusses this with the signature project of the class, computing the volume of the space between two perpendicular intersecting cylinders, and gets into how you go about visualizing such a problem.
Archimedes managed to find it, but only by virtue of his astounding ingenuity. He used a mechanical method based on levers and centers of gravity, in effect weighing the shape in his mind by balancing it against others he already understood. The downside of his approach, besides the prohibitive brilliance it required, was that it applied only to a limited range of shapes.
