The Quadrature of the Parabolic Segment

The problem of quadrature is one of finding the square or rectangle whose area is the same as that of a given region. In other words, it amounts to finding the area of the given region.Archimedes considered an arbitrary region bounded by an arc of a parabola, ADBEC, and a straight line, AC.

The point B is on the arc of the parabola where the tangent line is parallel to line AC,. D is the point where the tangent line is parallel to AB, and E is the point where the tangent line is parallel to BC. Archimedes proves that the areas of triangles ADB and BEC are each 1/8 of the area of trianble ABC. You can find this proof on pages 54–57 of Archimedes: What did he do besides cry Eureka? by Sherman Stein.