Example 7. Determine the area bounded by one arch of the cycloid $x=a(t-\sin t), y=a(1-\cos t)$ and the $O x$ axis (see Fig. 5.11).
Solution. By formula (5.14) we get:

$$
S=\int_{0}^{2 \pi a} y d x=\int_{0}^{2 \pi} a(1-\cos t) a(1-\cos t) d t=a^{2} \int_{0}^{2 \pi}(1-2 \cos t+
$$

$$
\begin{gathered}
+\cos ^{2} t) d t=a^{2} \int_{0}^{2 \pi}\left(1-2 \cos t+\frac{1+\cos 2 t}{2}\right) d t=a^{2} \int_{0}^{2 \pi}\left(\frac{3}{2}-2 \cos t+ \\
+\frac{1}{2} \cos 2 t\right) d t=\frac{3}{2} a^{2} \int_{0}^{2 \pi} d t-2 a^{2} \int_{0}^{2 \pi} \cos t d t+\frac{1}{4} a^{2} \int_{0}^{2 \pi} \cos 2 t d(2 t)= \\
=\left.\frac{3}{2} a^{2} t\right|_{0} ^{2 \pi}-\left.2 a^{2} \sin t\right|_{0} ^{2 \pi}+\left.\frac{a^{2}}{4} \sin 2 t\right|_{0} ^{2 \pi}=3 \pi a^{2}
\end{gathered}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-212.jpg?height=303&width=580&top_left_y=542&top_left_x=25)

Fig. 5.11