## Problem Statement

Calculate the area of the figure bounded by the lines given by the equations.

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=3(t-\sin t) \\
y=3(1-\cos t)
\end{array}\right. \\
& y=3(0<x<6 \pi, y \geq 3)
\end{aligned}
$$
## Solution

Let's find the points of intersection:

$$
\begin{aligned}
& y=3(1-\cos t)=3 \\
& 1-\cos t=1 \\
& \cos t=0 \\
& t=\frac{\pi}{2}+\pi n, n \in \mathbb{Z}
\end{aligned}
$$

We are interested in the interval \(0 < x < 6\pi\). Then the abscissas of the points of intersection will be:

![](https://cdn.mathpix.com/cropped/2024_05_22_fa23e2deef6f8d90c6d5g-17.jpg?height=1002&width=1014&top_left_y=1098&top_left_x=909)

$$
\begin{aligned}
& t=\frac{\pi}{2} \Rightarrow x=3\left(\frac{\pi}{2}-\sin \frac{\pi}{2}\right)=\frac{3 \pi}{2}-3 \\
& t=\frac{3 \pi}{2} \Rightarrow x=3\left(\frac{3 \pi}{2}-\sin \frac{3 \pi}{2}\right)=\frac{9 \pi}{2}+3
\end{aligned}
$$

Calculate the area:

$$
S=\int_{\pi / 2}^{3 \pi / 2} 3(1-\cos t) \cdot(3(t-\sin t))' d t=
$$

$$
\begin{aligned}
& =9 \cdot \int_{\pi / 2}^{3 \pi / 2}(1-\cos t) \cdot(1-\cos t) d t= \\
& =9 \cdot \int_{\pi / 2}^{3 \pi / 2}\left(1-2 \cos t+\cos ^{2} t\right) d t=\left.9 \cdot(t-2 \sin t)\right|_{\pi / 2} ^{3 \pi / 2}+9 \cdot \int_{\pi / 2}^{3 \pi / 2} \frac{1}{2}(1+\cos 2 t) d t= \\
& =9 \cdot\left(\frac{3 \pi}{2}-2 \sin \frac{3 \pi}{2}\right)-9\left(\frac{\pi}{2}-2 \sin \frac{\pi}{2}\right)+\left.\frac{9}{2} \cdot\left(t+\frac{1}{2} \sin 2 t\right)\right|_{\pi / 2} ^{3 \pi / 2}= \\
& =\frac{27 \pi}{2}+18-\frac{9 \pi}{2}+18+\frac{9}{2} \cdot\left(\frac{3 \pi}{2}+\frac{1}{2} \sin 3 \pi\right)-\frac{9}{2} \cdot\left(\frac{\pi}{2}+\frac{1}{2} \sin \pi\right)= \\
& =9 \pi+36+\frac{27 \pi}{4}+0-\frac{9 \pi}{4}-0=9 \pi+36+\frac{18 \pi}{4}=\frac{27 \pi}{2}+36
\end{aligned}
$$

Source — "http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+15-9$ "

Categories: Kuznetsov's Problem Book Integrals Problem 15 | Integrals

- Last modified: 07:17, 15 June 2010.
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## Problem Kuznetsov Integrals 15-10

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