A circle of radius r>0 rolls without slipping along the x–axis, generating one arch of a cycloid. The arch is traced by a point on the rim of the circle and is given parametrically for 0≤t≤2π by
x(t)=r(t−sint),y(t)=r(1−cost).
Let S be the region bounded above by the cycloid and below by the x–axis on this interval.
(a) Verify that the endpoints of the arch lie on the x–axis and determine their x–coordinates in terms of r. Conclude that S is a finite region.
(b) Write a definite integral that gives the exact area of S. Clearly indicate the limits of integration but do not evaluate yet.
(c) Evaluate the integral from part (b) and simplify the result completely, showing all steps.
(d) Interpret the numerical factor in your final answer in part (c) in the context of the geometry of the cycloid.