Solution
= Solution
Scale the <ellipsoid> to a unit <ball> using
$$
x=ar\sin\theta\cos\phi,\qquad y=br\sin\theta\sin\phi,\qquad z=cr\cos\theta,
$$
with $0\le r\le1$, $0\le\theta\le\pi$ and $0\le\phi<2\pi$. This is <spherical coordinates> followed by a diagonal <linear transformation>, so the <Jacobian determinant> is $abc\,r^2\sin\theta$. The required <volume integral> is
$$
\boxed{V=abc\int_0^{2\pi}\int_0^\pi\int_0^1r^2\sin\theta\,dr\,d\theta\,d\phi
=\frac{4\pi abc}{3}.}
$$