Example 2.81. Find the force of pressure that each side of a plate, having the shape of a semi-ellipse, experiences if it is submerged vertically in a liquid so that the major semi-axis is at the surface of the liquid.
Solution. From the equation of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$ for the arc located in the fourth quadrant

$$
y(x)=-\frac{b}{a} \sqrt{a^{2}-x^{2}}
$$

the element of area $d S=\frac{b}{a} \sqrt{a^{2}-x^{2}} d x$, and the element of pressure according to Pascal's law $d F=\rho g x d S(x)=\rho g x \frac{b}{a} \sqrt{a^{2}-x^{2}} d x$. Therefore,

$$
\begin{gathered}
F=\int_{x_{1}}^{x_{2}} \rho g x d S(x)=\int_{-a}^{a} \rho g x \frac{b}{a} \sqrt{a^{2}-x^{2}} d x= \\
=2 \int_{0}^{a} \rho g x \frac{b}{a} \sqrt{a^{2}-x^{2}} d x=-\left.g \rho \frac{2 b}{3 a}\left(a^{2}-x^{2}\right)^{3 / 2}\right|_{0} ^{a}=\frac{2 b a^{2}}{3} \rho g .
\end{gathered}
$$