Parabolic Segment

A parabolic segment is the region bounded by a parabola and a chord. For and , the arc length of the parabolic arc

illustrated above is given by

and the area is given by

(Kern and Bland 1948, p. 4). The first moment about the -axis is

so the geometric centroid is then given by

The area of the cut-off parabolic segment contained between the curves

for can be found by eliminating ,

so the points of intersection are

with corresponding -coordinates . The area is therefore given by

The maximum area of a triangle inscribed in this segment will have two of its polygon vertices at the intersections and , and the third at a point to be determined. From the general formula for triangle area, the area of the inscribed triangle is given by the determinant equation

Substituting with gives

To find the maximum area, differentiate with respect to and set the result to 0 to obtain

so

Substituting into the triangle area gives

This leads to the result known to Archimedes in the third century BC, namely