Example 2.65. Calculate the volume of the body formed by the rotation around the $O Y$ axis of the curvilinear trapezoid bounded by the hyperbola $x y=2$ and the lines $y_{1}=1, y_{2}=4$ and $y_{3}=0$.
Solution.

$$
V_{O Y}=\pi \int_{y_{1}}^{y_{2}}(x(y))^{2} d y=\pi \int_{0}^{4} \frac{4}{y^{2}} d y=4 \pi[-0.25+1]=3 \pi
$$

## 2.5 .  MECHANICAL AND PHYSICAL APPLICATIONS OF DEFINITE INTEGRALS