If you unwind thread from a stationary circular spool, keeping the thread taut at all times, then the endpoint traces a curve C called the involute of the circle (Figure 9). Observe that PQ has length Rθ. Show that C is parametrized by c(θ) = R(cos θ + θ sin θ), R(sin θ − θ cos θ) Then find the length of the involute for 0 ≤ θ ≤ 2π.

on R0 R Q 0 P = (x, y) P = (x, y) X