63. Prove a famous result of Archimedes (generalizing Exercise 62): For r < s, the area of the shaded region in Figure 9 is equal to four-thirds the area of triangle AACE, where C is the point on the parabola at which the tangent line is parallel to secant line AE. (a) Show that C has x-coordinate (r +s)/2. (b) Show that AB DE has area (s – r) /4 by viewing it as a parallelogram of height s – r and base of length CF. (c) Show that AACE has area (s – r) /8 by observing that it has the same base and height as the parallelo- gram. (d) Compute the shaded area as the area under the graph minus the area of a trapezoid, and prove Archimedes's result. FIGURE 9 Graph of f(x) = (x – a)(b – x).

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