Percentage Errors in Approximating the Volumes of a Wine Barrel and a Goblet

The German astronomer Johannes Kepler (1571–1630) found that the planets orbit the Sun in ellipses and showed that comets increase in speed as they near the Sun. He also estimated the volume of a wine barrel with height , base and top radii , and midsection radius as:
volume = .
In this Demonstration, assume that the revolving curve is a part of a circle with radius and center at a distance from the center of the barrel.
The exact volume is calculated using calculus and the percentage error is shown.
The goblet is constructed using only harmonic division.
The side of the square is divided into the three-term harmonic proportion: .
The radius of the goblet is one-half the side of the square. The volume of the goblet can be estimated by times the volume of a sphere of radius .
The percentage error in calculating the volume using this estimate compared to the exact value (using calculus) is very small.

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