Example. Find the area of the figure bounded by the given lines

$$
y^{2}-4 y+x^{2}=0, \quad y^{2}-8 y+x^{2}=0, \quad y=\frac{x}{\sqrt{3}}, \quad x=0
$$
## Solution.

1. Since the region $D$ is bounded by circles and lines passing through the origin, it is easier to solve the given problem by transitioning to polar coordinates

$$
\left\{\begin{array}{l}
x=\rho \cos \varphi \\
y=\rho \sin \varphi
\end{array}\right.
$$

In this case, the region $D$ will transform into the region $D^{\prime}$, bounded by the lines

$$
\varrho=4 \sin \varphi, \varrho=8 \sin \varphi, \varphi=\frac{\pi}{6}, \varphi=\frac{\pi}{2}
$$

And the desired area will be equal to

$$
S=\iint_{D^{\prime}} \varrho d \varrho d \varphi
$$

2. Define the region $D^{\prime}$ in polar coordinates using inequalities:

$$
D^{\prime}=\left\{\begin{array}{lc}
(\varrho, \varphi): & \pi / 6 \leq \varphi \leq \pi / 2 \\
4 \sin \varphi \leq \varrho \leq 8 \sin \varphi
\end{array}\right\}
$$

3. Transition from the double integral to the repeated integral:

$$
S=\int_{\pi / 6}^{\pi / 2} d \varphi \int_{4 \sin \varphi}^{8 \sin \varphi} \varrho d \varrho
$$

Integrating sequentially, we get

$$
S=\int_{\pi / 6}^{\pi / 2} d \varphi \int_{4 \sin \varphi}^{8 \sin \varphi} \varrho d \varrho=\left.\int_{\pi / 6}^{\pi / 2} \frac{\varrho^{2}}{2}\right|_{4 \sin \varphi} ^{8 \sin \varphi} d \varphi=24 \int_{\pi / 6}^{\pi / 2} \sin ^{2} \varphi d \varphi=4 \pi+3 \sqrt{3}
$$

Answer. $S=(4 \pi+3 \sqrt{3})(\text { units of length })^{2}$.

Conditions of the Problems. Find the areas of figures bounded by the given lines.

1. $y^{2}-3 y+x^{2}=0, \quad y^{2}-5 y+x^{2}=0, \quad y=x / \sqrt{3}, \quad x=0$.
2. $y^{2}+3 y+x^{2}=0, \quad y^{2}+5 y+x^{2}=0, \quad y=x, \quad x=0$.
3. $y^{2}-3 y+x^{2}=0, \quad y^{2}-7 y+x^{2}=0, \quad y=\sqrt{3} x, \quad x=0$.
4. $y^{2}+3 y+x^{2}=0, \quad y^{2}+5 y+x^{2}=0, \quad y=x, \quad y=x / \sqrt{3}$.
5. $y^{2}-3 y+x^{2}=0, \quad y^{2}-5 y+x^{2}=0, \quad y=x / \sqrt{3}, \quad x=\sqrt{3} x$.
6. $y^{2}-5 x+x^{2}=0, \quad y^{2}-7 x+x^{2}=0, \quad y=x / \sqrt{3}, \quad y=0$.
7. $y^{2}+4 x+x^{2}=0, \quad y^{2}+6 x+x^{2}=0, \quad y=x / \sqrt{3}, \quad y=\sqrt{3} x$.
8. $y^{2}-3 x+x^{2}=0, \quad y^{2}-5 x+x^{2}=0, \quad y=\sqrt{3} x, \quad y=0$.
9. $y^{2}+2 x+x^{2}=0, \quad y^{2}+4 x+x^{2}=0, \quad y=x, \quad y=0$.
10. $y^{2}+2 x+x^{2}=0, \quad y^{2}+10 x+x^{2}=0, \quad y=\sqrt{3} x, \quad y=0$.

Answers. 1. $S=4 \pi / 3+\sqrt{3} . \quad$ 2. $S=\pi+2 . \quad$ 3. $S=5 \pi / 3+5 \sqrt{3} / 2$. 4. $S=\pi / 3-2+\sqrt{3}$. $\quad 5 . S=2 \pi / 3$. $\quad 6 . S=\pi+3 \sqrt{3} / 2$. $\quad$ 7. $S=5 \pi / 6$. 8. $S=4 \pi / 3+\sqrt{3}$. $\quad 9$. $S=3 \pi / 4+3 / 2$. $\quad 10 . S=8 \pi+6 \sqrt{3}$.

## 12.8. Calculation of the Mass of a Flat Plate

Problem Statement. Find the mass of a flat plate $D$ with surface density $\mu=\mu(x, y)$, bounded by the given curves.

Plan of Solution.

1. The mass of the plate $D$ with surface density $\mu(x, y)$ is determined by the formula

$$
m=\iint_{D} \mu(x, y) d x d y
$$

2. Compute the resulting double integral. Write the answer, not forgetting the units of measurement.