Short answers, pulled from the story.
The Moscow Mathematical Papyrus dates to around 1820 BCE and contains early volume calculations. Ancient Egyptian scribes used natural containers to measure grain and liquids in the Reisner Papyrus.
Archimedes lived during the third century BCE and devised approximate volume formulas using the method of exhaustion. He derived solutions from previous known formulas regarding similar shapes.
French law formally defined the metric system on the 7th of April 1795 using six specific units. Three units related directly to volume: the stère for firewood, the litre for liquids, and the gramme for mass.
One thousand cubic centimetres equal one cubic decimetre. The litre serves as another metric unit where one litre equals one cubic decimetre or 1000 cubic centimetres.
Integral calculus calculates volumes of solids of revolution by rotating plane curves around lines on the same plane. The washer or disc integration method integrates by an axis parallel to the rotation axis while shell integration methods integrate when axes are perpendicular to the rotation axis.