On the area of a parabolic sector

I have developed the habit, while in my department’s copy room waiting for the machine to churn out handouts, quizzes, and assignment sheets, of browsing through the excellent collection A Century of Calculus ([1]) that is kept on the shelves there. So it happened that I came across the article Area of a parabolic region, by Rozen and Sofo ([3]), wherein the authors show that the area of a region bounded by a parabola and a chord that is perpendicular to the parabola’s axis of symmetry is equal to 2/3 of the product of the length of the chord and the height of the region. This is, of course, a special case of a result of Archimedes that the area of any parabolic sector is given by 4/3 of the area of the inscribed triangle defined by the endpoints of the chord and the point on the parabola where the tangent line is parallel to the chord, or, equivalently, by 2/3 of the area of the circumscribed parallelogram. Nonetheless, the “base times height” flavor of Rozen and Sofo’s result appealed to me, and I wondered how to express the general result in those familiar terms. (See [2] for the authoritative work on Archimedes and [4] for a modern treatment of some of his results on the parabola.)