Example 2.53. Compute the area bounded by the cardioid $\rho=1+\cos \varphi,(0 \leqslant \varphi \leqslant 2 \pi)$.
Solution. The cardioid is depicted in Fig. 2.7. We will use formula (2.23):

$$
\begin{aligned}
S & =0.5 \int_{0}^{2 \pi}(\rho(\varphi))^{2} d \varphi=0.5 \int_{0}^{2 \pi}(1+\cos \varphi)^{2} d \varphi= \\
& =0.5 \int_{0}^{2 \pi}\left(1+2 \cos \varphi+\cos ^{2} \varphi\right) d \varphi= \\
& =0.5 \int_{0}^{2 \pi}(1+2 \cos \varphi+0.5(1+\cos 2 \varphi)) d \varphi= \\
& =0.5\left(1.5 \varphi+2 \sin \varphi+0.25 \sin 2 \varphi\right)\bigg|_{0}^{2 \pi}=\frac{3 \pi}{2}
\end{aligned}
$$