Parabolic Segment

The arc length of the parabolic segment

illustrated above is given by

and the area is given by

(Kern and Bland 1948, p. 4). The weighted mean of is

so the geometric centroid is then given by

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

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 equation for a triangle, the area of the inscribed triangle is given by the determinant equation

Plugging in and using gives

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

so

Plugging (21) into (19) then gives

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