Volume Of A Sphere
Hemispheres can be approximated by stacks of cylinders:
For a hemisphere of radius r, N cylinders would each have a height of approximately r / N. The set of approximate cylinder radii would be √r2 - (r / N)2, √r2 - (2r / N)2, √r2 - (3r / N)2, ..., 0. The approximate volume of all the cylinders would be πr3- (πr3 / N3)(1 + 4 + 9 + ... + N2). As N increases, the volume approximation improves and (1 + 4 + 9 + ... + N2) / N3 approaches 1 / 3. Therefore, the volume of a sphere of radius r is 4πr3 / 3.
For a hemisphere of radius r, N cylinders would each have a height of approximately r / N. The set of approximate cylinder radii would be √r2 - (r / N)2, √r2 - (2r / N)2, √r2 - (3r / N)2, ..., 0. The approximate volume of all the cylinders would be πr3- (πr3 / N3)(1 + 4 + 9 + ... + N2). As N increases, the volume approximation improves and (1 + 4 + 9 + ... + N2) / N3 approaches 1 / 3. Therefore, the volume of a sphere of radius r is 4πr3 / 3.
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