A barrel solid of revolution composed of parallel circular top and bottom with a common axis and a side formed by a smooth curve symmetrical
about the midplane.

The term also has a technical meaning in functional analysis. In particular, a subset of a topological linear space is a barrel if
it is absorbing, closed, and absolutely convex (Taylor and Lay 1980, p. 111).
(A subset of a topological linear space is absorbing if for each there is an such that is in
if for each
such that . A subset of a topological linear space is absolutely convex if for
each and in ,
is in if .)

When buying supplies for his second wedding, the great astronomer Johannes Kepler became unhappy about the inexact methods used by the merchants to estimate the liquid
contents of a wine barrel. Kepler therefore investigated the properties of nearly
100 solids of revolution generated by rotation
of conic sections about non-principal axes (Kepler, MacDonnell, Shechter, Tikhomirov
1991).

For sides consisting of an arc of an ellipse, the equation
of the side is given by