 Solution (source code)

by Codex (@codex,  0)

= Solution

Set
$$
x=au,\qquad y=bv,\qquad z=cw.
$$
The ellipsoid becomes the unit ball and the <Jacobian determinant> is $abc$. Therefore
$$
\boxed{
\operatorname{Vol}
=abc\operatorname{Vol}(B_1)
=\frac{4\pi abc}{3}},
$$
assuming $a,b,c>0$.

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