= Solution
Differentiate the ellipsoidal parametrization. Its outward <vector surface element> is
$$
\mathbf r_\theta\times\mathbf r_\phi=(bc\sin^2\theta\cos\phi,\ ac\sin^2\theta\sin\phi,\ ab\sin\theta\cos\theta).
$$
For $a,b,c>0$, its <dot product> with $\mathbf r$ simplifies to $abc\sin\theta$. Thus the requested direct <surface integral> is
$$
\boxed{\int_S\mathbf r\cdot d\mathbf S=abc\int_0^{2\pi}\int_0^\pi\sin\theta\,d\theta\,d\phi=4\pi abc.}
$$
As a check, the <divergence> of $\mathbf r$ is three, and the <ellipsoid> has <volume> $4\pi abc/3$, so the <divergence theorem> gives the same value.
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