The arc length L of a curve given parametrically by

(x(t), y(t)) for a ≤ t ≤ b

is given by the formula

L = integral a to b(x '(t))2 + (y '(t))2dt

A path of a point on the edge of a rolling circle of radius R is a cycloid, given by

x(t) = R (t − sin t),


y(t) = R (1 − cos t),

where t is the angle (in radians) the circle has rotated.



Find the length L of one "arch" of this cycloid, that is, find the distance traveled by a small stone stuck in the tread of a tire of radius R during one revolution of the rolling tire.