884. Find the length of the cardioid $x=2 a \cos t-a \cos 2 t, y=$ $-2 a \sin t-a \sin 2 t$.
Solution. We apply formula (1); based on the given parametric equations of the cardioid and the formula for the differential of the arc of a plane curve (Chapter 5, §), we transform the line integral in formula (1) into an ordinary integral with the variable $t$.

$$
\begin{gathered}
\dot{x}=-2 a \sin t+2 a \sin 2 t, \quad \dot{y}=2 a \cos t-2 a \cos 2 t \\
d l=\sqrt{x^{2}+y^{2}} d t=4 a \sin \frac{t}{2} d t
\end{gathered}
$$

The cardioid (Fig. 187) is obtained as $t$ varies from 0 to $2 \pi$. Therefore,

$$
L=4 a \int_{0}^{2 \pi} \sin \frac{t}{2} d t=-\left.8 a \cos \frac{t}{2}\right|_{0} ^{2 \pi}=16 a
$$