1325. The boiler has the shape of a paraboloid of revolution (Fig. 92). The radius of its base $R=3$ m, depth $H=5$ m. The boiler is filled with a liquid of specific weight $0.8 \Gamma /$ cm $^{3}$. Calculate the work required to pump the liquid out of the boiler.
Solution. In the section plane $x O y A O B$, there is a parabola with the equation $y=a x^{2}$. Let's find the parameter $a$. The coordinates of point $B$ must satisfy this equation, i.e., $H=a R^{2}$, hence $a=\frac{H}{R^{2}}$, so $y=\frac{H}{R^{2}} x^{2}$.

Divide the paraboloid into layers by planes parallel to the surface of the liquid. Let the thickness of the layer at depth $H-y$ be $d y$. Then, approximating the layer as a cylinder, we get its volume

$$
d V=\pi x^{2} d y
$$

From the equation of the parabola, we find $x^{2}=\frac{y R^{2}}{H}$. Substituting this value into formula (a), we get

$$
d V=\frac{\pi y R^{2}}{H} d y
$$

i.e., the weight of the liquid layer is $\gamma d V=\frac{\tau y R^{2} \gamma}{H} d y$. Therefore, to pump the liquid from depth $H-y$, the elementary work required is

$$
d A=\frac{\pi y R^{2} \gamma(H-y)}{H} d y
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_1b9a1c1720bae4af5bc5g-259.jpg?height=395&width=532&top_left_y=1377&top_left_x=585)

Fig. 92

Considering that $0 \leqslant y \leqslant H$, we get

$$
\begin{gathered}
A=\int_{0}^{H} \frac{\pi y R^{2} \gamma(H-y)}{H} d y=\frac{\pi R^{2} \gamma}{H} \int_{0}^{H} y(H-y) d y= \\
=\left.\frac{\pi R^{2} \gamma}{H}\left(\frac{H y^{2}}{2}-\frac{y^{3}}{3}\right)\right|_{0} ^{H}=\frac{\pi R^{2} \gamma}{H}\left(\frac{H^{3}}{2}-\frac{H^{3}}{3}\right)=\frac{1}{6} \pi R^{2} \gamma H^{2} .
\end{gathered}
$$

Substituting the values of $R$ and $H$, expressed in meters, and $\gamma=$ $=800 \kappa \Gamma$, we find the required work

$$
A=\frac{1}{6} \pi 3^{2} \cdot 800 \cdot 5^{2}=30000 \pi(\kappa \Gamma \mathcal{M}) \approx 30000 \cdot 9.81 \pi(\partial \varkappa)=
$$

$=294300 \pi(\partial \varkappa)$ (J (joule) - the unit of work in the International System of Units; $1 \kappa \Gamma m \approx 9.81$ J).