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×2^1/2 × (1-cos(t))^1/2.

Using the hint:

r×2^1/2 × 2^1/2 × integral of sin(t/2) dt from 0 to 2×pi

= 2×r × (-2×cos(t/w) from 2×pi to 0)
= 8r

Reference: Example 3, page 550, Calculus and Analytic Geometry, 1982 edition.

Michael Shackleford, A.S.A.