Example 2.59. Calculate the length of the arc of the cardioid $\rho=$ $=2(1+\cos \varphi), 0 \leqslant \varphi \leqslant 2 \pi$ (see Fig. 2.7).
Solution. We will use formula (2.28):

$$
\begin{aligned}
L & =\int_{0}^{2 \pi} \sqrt{(2(1+\cos \varphi))^{2}+(-2 \sin \varphi)^{2}} d \varphi=\int_{0}^{2 \pi} \sqrt{16 \cos ^{2}(\varphi / 2)} d \varphi= \\
& =4 \int_{0}^{2 \pi}|\cos (\varphi / 2)| d \varphi=4 \cdot 2 \int_{0}^{\pi} \cos (\varphi / 2) d \varphi=\left.16 \sin (\varphi / 2)\right|_{0} ^{\pi}=16
\end{aligned}
$$

(due to the properties of the function $\cos (\varphi / 2)$ ).