Chapter 6, Problem 86RE

Equal area property for parabolas Let f(x) = ax² + bx + c be an arbitrary quadratic function and choose two points x = p and x= q. Let L₁ be the line tangent to the graph of f at the point (p, f(p)) and let L₂ be the line tangent to the graph at the point (q, f(q)). Let x = s be the vertical line through the intersection point of L₁ and L₂. Finally, let R₁, be the region bounded by y =f(x), L₁, and the vertical line x = s, and let R₂ be the region bounded by y = f(x), L₂, and the vertical line x = s. Prove that the area of R₁ equals the area of R₂.