Example 4. Calculate the volume of the body obtained by rotating the ellipse

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-221.jpg?height=412&width=602&top_left_y=1078&top_left_x=43)

Fig. 5.22

$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
$$

around the $O y$ axis. (This body is called an ellipsoid of revolution, see Fig. 5.22.)
Solution. Since in this case $c=-b, d=b$,

$$
x^{2}=a^{2}\left(1-\frac{y^{2}}{b^{2}}\right)
$$

then by formula (5.22) we determine

$$
V_{y}=\pi \int_{-b}^{b} a^{2}\left(1-\frac{y^{2}}{b^{2}}\right) d y=\pi a^{2} \int_{-b}^{b} d y-\pi \frac{a^{2}}{b^{2}} \int_{-b}^{b} y^{2} d y=
$$

$$
\begin{gathered}
=\left.\pi a^{2} y\right|_{-b} ^{b}-\left.\frac{\pi a^{2}}{3 b^{2}} y^{3}\right|_{-b} ^{b}=\pi a^{2}[b-(-b)]-\frac{\pi a^{2}}{3 b^{2}}\left[b^{3}-(-b)^{3}\right]= \\
=2 \pi a^{2} b-\frac{2}{3} \pi a^{2} b=\frac{4}{3} \pi a^{2} b
\end{gathered}
$$

Thus, $V=\frac{4}{3} \pi a^{2} b$. When $a=b=R$, we get $V=\frac{4}{3} \pi R^{3}$ - the volume of a sphere.