P.o.t.W. #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.

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