1307. An ellipse with a major axis of $2a$ and a minor axis of $2b$ ($a > b$) rotates: 1) around the major axis; 2) around the minor axis. Find the volumes of the resulting ellipsoids of revolution. In a special case, determine the volume of a sphere.
Solution. Let's write the equation of the ellipse:

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

Using formula (1), we will find the volume of the body formed by rotating the ellipse around the $O x$ axis. From the equation of the ellipse,

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

By the condition, the major semi-axis of the ellipse is $a$, so the interval of integration will be from $-a$ to $a$ (Fig. 87):

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

from which

$$
V=\frac{4}{3} \pi a b^{2}
$$

Now, let's find the volume of the body formed by rotating the ellipse around the $O y$ axis. We will use formula (2).

From the equation of the ellipse,

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

By the condition, the minor semi-axis of the ellipse is $b$, so the interval of integration will be from $-b$ to $b$:

$$
\frac{1}{2} V=\pi \int_{0}^{b} \frac{a^{2}}{b^{2}}\left(b^{2}-y^{2}\right) d y=\frac{\pi a^{2}}{b^{2}} \int_{0}^{b}\left(b^{2}-y^{2}\right) d y=\frac{2}{3} \pi a^{2} b
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_1b9a1c1720bae4af5bc5g-254.jpg?height=326&width=429&top_left_y=703&top_left_x=68)

Fig. 87

![](https://cdn.mathpix.com/cropped/2024_05_22_1b9a1c1720bae4af5bc5g-254.jpg?height=488&width=474&top_left_y=556&top_left_x=629)

Fig. 88

from which

$$
V=\frac{4}{3} \pi a^{2} b
$$

A special case of the ellipsoid of revolution is when $a=b$, which is a sphere. Thus, the volume of the sphere $V=\frac{4}{3} \pi a^{3}$, where $a$ is the radius of the sphere.