In the following figure, arc ABC is a semicircle with

Chapter 9, Problem 49

(choose chapter or problem)

In the following figure, arc ABC is a semicircle with diameter and arc CDE is a semicircle with diameter (continues) E +=1 x y C D A B AC, CE.(a) Show that the area of semicircle CDE is p(1 cos u)/4.Hint: Use the chord-length formula in Exercise 47.(b) Express the area of lune CDE in terms of u.Hint: Use the result in part (a) along with the formulain Exercise 44(a) for the area of a segment.(c) Express the area of lune ABC in terms of u.(d) Express the area of ^ACE in terms of u.(e) Use the results in parts (b), (c), and (d) to verify thatthe area of ^ACE is equal to the sum of the areas ofthe two lunes CDE and ABC Remark: As with the results in Exercises 45 and 46, this resultabout lunes was discovered and proved by the ancient Greekmathematician Hippocrates of Chios. According to ProfessorGeorge F. Simmons in his book Calculus Gems (New York:McGraw-Hill Book Co., 1992), these results appear to be theearliest precise determination of the area of a region boundedby curves.