Answer
We prove that
${\rm{Volume}}\left( {\cal W} \right) = abc \times {\rm{Volume}}\left( {{{\cal W}_0}} \right)$,
where ${\cal W}$ and ${{\cal W}_0}$ denote the ellipsoid and the unit sphere, respectively.
${\rm{Volume}}\left( {\cal W} \right) = \frac{4}{3}\pi abc$
Work Step by Step
Write $x = au$, $y = bv$, and $z = cw$. So,
$u = \frac{x}{a}$, ${\ \ \ \ }$ $v = \frac{y}{b}$, ${\ \ \ \ }$ $w = \frac{z}{c}$
Then the interior of the ellipsoid ${\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} + {\left( {\frac{z}{c}} \right)^2} \le 1$ corresponds to the disk ${u^2} + {v^2} + {w^2} \le 1$ in the $\left( {u,v,w} \right)$-space. Thus, we can use $G\left( {u,v,w} \right) = \left( {au,bv,cw} \right)$ to map the unit sphere ${u^2} + {v^2} + {w^2} \le 1$ onto the interior of the ellipsoid ${\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} + {\left( {\frac{z}{c}} \right)^2} \le 1$.
Evaluate the Jacobian of $G\left( {u,v,w} \right) = \left( {au,bv,cw} \right)$:
${\rm{Jac}}\left( G \right) = \left| {\begin{array}{*{20}{c}}
{\frac{{\partial x}}{{\partial u}}}&{\frac{{\partial x}}{{\partial v}}}&{\frac{{\partial x}}{{\partial w}}}\\
{\frac{{\partial y}}{{\partial u}}}&{\frac{{\partial y}}{{\partial v}}}&{\frac{{\partial y}}{{\partial w}}}\\
{\frac{{\partial z}}{{\partial u}}}&{\frac{{\partial z}}{{\partial v}}}&{\frac{{\partial z}}{{\partial w}}}
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
a&0&0\\
0&b&0\\
0&0&c
\end{array}} \right| = abc$
Let ${{\cal W}_0}$ denote the region of the unit sphere and ${\cal W}$ denote the region of the ellipsoid. Using the Change of Variables Formula, we evaluate the volume of ${\cal W}$:
${\rm{Volume}}\left( {\cal W} \right) = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} {\rm{d}}x{\rm{d}}y{\rm{d}}z = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{{{\cal W}_0}}^{} \left| {Jac\left( G \right)} \right|{\rm{d}}u{\rm{d}}v{\rm{d}}w$
$ = abc\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{{{\cal W}_0}}^{} {\rm{d}}u{\rm{d}}v{\rm{d}}w$
Since ${\rm{Volume}}\left( {{{\cal W}_0}} \right) = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{{{\cal W}_0}}^{} {\rm{d}}u{\rm{d}}v{\rm{d}}w$, so
${\rm{Volume}}\left( {\cal W} \right) = abc \times {\rm{Volume}}\left( {{{\cal W}_0}} \right)$
Hence, the volume of the ellipsoid ${\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} + {\left( {\frac{z}{c}} \right)^2} = 1$ is equal to $abc$ $ \times $ the volume of the unit sphere.
The volume of the unit sphere is $\frac{4}{3}\pi $, so ${\rm{Volume}}\left( {\cal W} \right) = \frac{4}{3}\pi abc$.
