Example 1. Find the volume of the body obtained by rotating around the $O x$ axis the curvilinear trapezoid bounded by the parabola $y^{2}=2 x$, the line $x=3$ and the $O x$ axis.
Solution. According to the condition of the problem, we determine the limits of integration $a=0, b=3$ (Fig. 5.20).

By formula (5.21), we find

$$
V_{x}=\pi \int_{0}^{3} y^{2} d x=\pi \int_{0}^{3} 2 x d x=\left.\pi x^{2}\right|_{0} ^{3}=9 \pi
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-220.jpg?height=412&width=489&top_left_y=602&top_left_x=53)

Fig. 5.20

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-220.jpg?height=413&width=491&top_left_y=603&top_left_x=627)

Fig. 5.21