Archimedes, sometimes described as the inventor of integral calculus, is credited with
determining a theorem & formula to find the area enclosed by a chord of a
parabola.

This page:
 Area enclosed by a parabola
Area enclosed by chord of parabola
Formula for thea area enclosed by the chord of a parabola
Area = 2/3 area of circumscribed parallelogram formed by the
chord of the parabola and a tangent of
the parabola. See the picture below