Exercise on Proposition 4 of The Method

Investigate Archimedes' Proposition 4 below, which concerns a paraboloid (the solid obtained by rotating the curve x = y² about the x-axis) inscribed inside a cylinder.  A static version of the balancing appears below.  If you click on Proposition 4, you will get a dynamic version.

a) Describe the balancing relationship that is illustrated in the sketch.

b) Use this relationship to determine a relationship between the volume of the paraboloid and the volume of the cylinder.

c) Find a formula for the volume of the paraboloid in terms of r, the radius of the cylinder.

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