Example 2.64. Calculate the volume of the body obtained by rotating the part of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ over the segment $0 \leqslant x \leqslant a$ around the $O X$ axis.
Solution. We will use formula (2.35).

$$
\begin{gathered}
V_{O X}=\pi \int_{x_{1}}^{x_{2}} \frac{b^{2}}{a^{2}}\left(a^{2}-x^{2}\right) d x=\pi \frac{b^{2}}{a^{2}} \int_{0}^{a}\left(a^{2}-x^{2}\right) d x= \\
\quad=\left.\pi \frac{b^{2}}{a^{2}}\left(a^{2} x-\frac{x^{3}}{3}\right)\right|_{0} ^{a}=\frac{2 \pi}{3} a b^{2}
\end{gathered}
$$