Problem 3 Solution

Let t be the distance the center of the wheel has moved from (0,0). Then:

x=t+sin(t)
y=cos(t)

Taking the derivitives:

dx/dt=1+cos(t)
dy/dt=-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:

2^1/2 * (1+cos(t))^1/2.

Multiply by (1-cos(t))^1/2 / (1-cos(t))^1/2 and the integral becomes:

2^1/2 * sin(t) / (1-cos(t))^1/2.

Let u=cos(t) and the integral becomes:

2^1/2 * (1-u)^-1/2.

Integrating this you get:

2^1/2 * 2 * (1-u)^1/2 .

The bounds are 0 to 2*pi, so the total arc length is:

2^1/2 * 2 * (2^1/2+2^1/2) =

2^1/2 * 2 * 2 * 2^1/2 = 8