Answer
$$
f(x)=\frac{b}{a} \sqrt{a^{2}-x^{2}}
$$
The volume of an ellipsoid is given by :
$$
\begin{aligned} V &=\pi \int_{-a}^{a}(f(x))^{2} d x\\
&=\pi \int_{-a}^{a}\left[\frac{b}{a} \sqrt{a^{2}-x^{2}}\right]^{2} d x \\
&=\frac{4 a b^{2} \pi}{3}
\end{aligned}
$$
Work Step by Step
$$
f(x)=\frac{b}{a} \sqrt{a^{2}-x^{2}}
$$
The volume of an ellipsoid is given by :
$$
\begin{aligned} V &=\pi \int_{-a}^{a}(f(x))^{2} d x\\
&=\pi \int_{-a}^{a}\left[\frac{b}{a} \sqrt{a^{2}-x^{2}}\right]^{2} d x \\ &=\pi \int_{-a}^{a} \frac{b^{2}}{a^{2}}\left(a^{2}-x^{2}\right) d x \\ &=\left.\frac{\pi b^{2}}{a^{2}}\left(a^{2} x-\frac{x^{3}}{3}\right)\right|_{-a} ^{a} \\
&=\frac{\pi b^{2}}{a^{2}}\left[\left(a^{3}-\frac{a^{3}}{3}\right)-\left(-a^{3}+\frac{a^{3}}{3}\right)\right] \\
& =\frac{\pi b^{2}}{a^{2}}\left(2 a^{3}-\frac{2 a^{3}}{3}\right)\\
& =\frac{\pi b^{2}}{a^{2}}\left(\frac{4 a^{3}}{3}\right)\\
&=\frac{4 a b^{2} \pi}{3}
\end{aligned}
$$
