Ta có

$V=\pi \int\limits_{-a}^{a}{{{y}^{2}}dx}=2\pi \int\limits_{0}^{a}{\frac{{{b}^{2}}}{{{a}^{2}}}\left( {{a}^{2}}-{{x}^{2}} \right)dx}=\frac{2\pi {{b}^{2}}}{{{a}^{2}}}\left( {{a}^{2}}x-\frac{{{x}^{3}}}{3} \right)\left| \begin{align}  & \pi  \\  & 0 \\ \end{align} \right.$

$=\frac{2\pi {{b}^{2}}}{{{a}^{2}}}\left( {{a}^{3}}-\frac{{{a}^{3}}}{3} \right)=\frac{4}{3}\pi a{{b}^{2}}$