Mathematical Discoveries - The Quadrature of A Parabolic Segment

The area of a parabolic segment:

 

 

     As proven by Archimedes, the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure.
 

     In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is ⁴⁄₃ times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio ¹⁄₄:

 

     If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and so on. This proof uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 +……. which sums to ¹⁄₃.