Example 6. Find the coordinates of the center of gravity of a homogeneous plane figure bounded by the lines $y=\frac{1}{2} x^{2} ; y=2$.
Solution. Since the figure is homogeneous (we assume $\rho(x, y)=1$), then due to its symmetry relative to the $O y$ axis (Fig. 3.15), the abscissa of the center of gravity $\bar{x}=0$. For the ordinate, we have:
![](https://cdn.mathpix.com/cropped/2024_05_22_446938f1e4808e7d6e07g-059.jpg?height=358&width=894&top_left_y=393&top_left_x=126)

Calculation of the integrals, taking into account the symmetry relative to the $O y$ axis, gives:

Fig. 3.15

$$
\begin{gathered}
\iint_{D} d x d y=2 \int_{0}^{2} d x \int_{x^{2} / 2}^{2} d y=2 \int_{0}^{2}\left(2-\frac{x^{2}}{2}\right) d x=\frac{16}{3} \\
\iint_{D} y d x d y=2 \int_{0}^{2} d x \int_{x^{2} / 2}^{2} y d y=2 \int_{0}^{2}\left(2-\frac{x^{4}}{8}\right) d x=\frac{32}{5}
\end{gathered}
$$

Therefore, $\bar{y}=\frac{6}{5}$.

Answer. $\left(0, \frac{6}{5}\right)$

## Exercises

Calculate the areas of the surfaces (1-4).

1. Parabolic cylinder $2 z=x^{2}$ between the planes $y=\frac{x}{2}$, $y=2 x$ and $x=2 \sqrt{2}$.
2. Cone $y^{2}+z^{2}=x^{2}$ inside the cylinder $x^{2}+y^{2}=a^{2}$
3. Paraboloid $x^{2}+y^{2}=2 a z$ inside the cylinder $x^{2}+y^{2}=3 a^{2}$.
4. Paraboloid $z=x^{2}-y^{2}$ between the paraboloids $z=3 x^{2}+y^{2}-2$ and $z=3 x^{2}+y^{2}-4$.
5. Calculate the mass of the plate $D$ with density $\rho(x, y)$, if $D$:

a) triangle $A B C, A(0,0), B(1,1), C(0,1), \rho=y$;

b) $y^{2}-x^{2} \leqslant a^{2}, 0 \leqslant x \leqslant a, y \geqslant 0, \rho=\sqrt{a^{2}+x^{2}}$;

c) $x \geqslant 0, e^{x} \leqslant y \leqslant 2, \rho=e^{x+y}$.
6. Calculate the mass of the upper half of the sphere $x^{2}+y^{2}+z^{2}=R^{2}$, with density $\rho=z$.

Calculate the moments of inertia of the plate bounded by the given curves relative to the origin.
7. $x y=1, x y=2, y=x, y=2 x \quad(x>0, y>0)$. Hint. Make the substitution $x y=u, \frac{y}{x}=v$.
8. $x^{2}+y^{2}=\stackrel{x}{a} x, x^{2}+y^{2}=2 a x, y=0, y>0$.
9. $y=x, x+y=2 a, x=0$.

Calculate the coordinates of the center of gravity of the plate bounded by the given curves with density $\rho=1(10-11)$.
10. $a y=x^{2}, x+y=2 a \quad(a>0)$.
11. $\left(x^{2}+y^{2}\right)^{2