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Kepler's barrel rule treats the staves as parabolic arcs and gives V = πh(2D² + d²)/12, where D is the diameter at the widest point and d at the ends.
Kepler barrel rule
V = πh(2D² + d²) / 12
The formula becomes πhD²·3/12 = πD²h/4, the plain cylinder, as expected.
For real barrels with gently curved staves it is within a percent or so; a parabolic profile makes it exact.