Problem of the Week #9

■  No GDC  ■

Consider a parabola with equation \(y = a{x^2}\) such that \(a > 0\). Such a parabola is intersected by a line at two distinct points having coordinates \(\left( {b,a{b^2}} \right)\) and \(\left( {c,a{c^2}} \right)\) where \(b < c\) as shown in the figure below.

Show that the area bounded by the parabola and the line, a parabolic segment (shaded region), is equal to \(\frac{a}{6}{\left( {c - b} \right)^3}\).  Comment on the result.