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10 11 cavalIerIs PrIncIPle a proof by approximation in slices There are two versions of the celebrated principle named after Bonaventura Cavalieri 1598 1647. For plane figures it says that if every horisontal line intersects two plane figures in cuts of equal length, then the two figures will have equal area. Similarly, if every horisontal plane intersects two solids in cuts of equal area, then the two solids will have equal volume. An outline of the proof by approximation in slices, which is the same for both principles, is given on the opposite page. Cavalieris principle is a good example of divide into manageable pieces and conquer in mathematics. For example, in the plane version, we reduce the difficult problem of calculating areas to the easier problem of measuring the lengths of line segments. Below are some important area and volume formul easily derived using Cavalieris principle.
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