Oh boy! Here is a quote from the introduction to the work of archimedes where it states explicitly, that archimedes found a volume of a certain section of a cylinder, by reducing it to the problem of the area of a parabola.
"Proposition XI is the interesting case of a segment of a right
cylinder cut off by a plane through the center of the lower base and
tangent to the upper one. He shows this to equal onesixth of the
square prism that circumscribes the cylinder. This is well known to us
through the formula $v = 2r^2h/3$, the volume of the prism being
$4r^2h$, and requires a knowledge of the center of gravity of the
cylindric section in question. Archimedes is, so far as we know, the
first to state this result, and he obtains it by his usual method of
the skilful balancing of sections. There are several lacunae in the
demonstration, but enough of it remains to show the ingenuity of the
general plan. The culminating interest from the mathematical
standpoint lies in proposition XIII, where Archimedes reduces the
whole question to that of the quadrature of the parabola."
By the way, the famous work of Galileo in the 1600's of discovering that a moving projectile travels in the path of a parabola, and that the distances traveled by a falloing object, in succeeding units of time, stand to one another as the squares of the positibe integers, are also mathematical consequences of the work of archimedes.
this causes one to wonder why they were thought to be new in galileo's time, and why a genius like galileo did not realize they were corollaries of archimedes work.
of course the connection of the mathematics with the physics is in itself a significant discovery, but galileo seems to rederive all the mathematics by geometry. this puzzles me.
