Cavalieris proof of the volume formula for the sphere compares the half sphere
of radius L with the complement of a cone in a cylinder of radius L and height L.
The cross section of each body at height z has area (L2-z2)1/2.
Since the cylinder has volume pi L3 and the cone has a third of this volume,
the hemi sphere has 2/3 of the volume of the cylinder.
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|Oliver Knill, Math21a, Multivariable Calculus, Fall 2005, Department of Mathematics, Faculty of Art and Sciences, Harvard University|