Problem 3 Solution
Imagine the circle resting on coordinate (0,0) and moving east.
Also magine the point on the circle to start at (0,0).
Let t be the angle between the point on the circle and the center of the circle.
The position of the point on the circle, relative to t, is:
x = rt - r×sin(t)
y = r - r×cos(t)
Taking the derivatives:
dx/dt = r - r×cos(t)
dy/dt = r×sin(t)
The change in arc length can be defined as
( (dx/dt)2 + (dy/dt)2 ) 1/2.
So the total arc length is the integral from 0 to 2pi of ( (dx/dt)2 + (dy/dt)2 ) 1/2.
After a few steps this integral becomes:
r×21/2 × (1-cos(t))1/2.Michael Shackleford, A.S.A.
Using the hint:
r×21/2 × 21/2 × integral of sin(t/2) dt from 0 to 2×pi
= 2×r × (-2×cos(t/w) from 2×pi to 0)
Reference: Example 3, page 550, Calculus and Analytic Geometry, 1982 edition.