Example 4. Find the length of the arc of the involute (unwinding) of a circle: $x=a(t \sin t+\cos t), y=a(\sin t-t \cos t)$ from point $A(t=0)$ to an arbitrary point $M(t)$.
Solution. By formula (5.19) we get

$$
l=\int_{0}^{t} \sqrt{x'^2+y'^2} d t=a \int_{0}^{t} \sqrt{(t \cos t)^{2}+(t \sin t)^{2}} d t=a \int_{0}^{t} t d t=\frac{a t^{2}}{2}
$$

The cycloid is depicted in Fig. 5.14.