Find the volume of the solid bounded by the ellipsoid

Find the volume of the solid bounded by the ellipsoid: $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1. $$

Explanation of Method

The substitution $u=x/a,\,v=y/b,\,w=z/c$ maps the ellipsoid to the unit sphere, with a constant Jacobian $abc$; the volume follows immediately from the known volume of a sphere.

Substitute to reduce the ellipsoid to a unit sphere.
Compute the (constant) Jacobian.
Scale the unit sphere's volume by the Jacobian.

Step 1 · Substitute

Let $u=x/a,\,v=y/b,\,w=z/c$; the ellipsoid becomes $u^2+v^2+w^2=1$, the unit sphere.

Step 2 · Compute the Jacobian

With $x=au,\,y=bv,\,z=cw$: $\dfrac{\partial(x,y,z)}{\partial(u,v,w)}=abc$, so $dx\,dy\,dz=abc\,du\,dv\,dw$.

Step 3 · Scale by the sphere's volume

$$ V = abc\iiint_{u^2+v^2+w^2\leqslant1}du\,dv\,dw = abc\cdot\frac43\pi = \frac{4\pi abc}3. $$

Additional Notes

This is the classical formula for the volume of an ellipsoid, matching the source exactly (confirmed independently via slicing perpendicular to the $z$-axis, where each cross-section is an ellipse of area $\pi ab\left(1-\frac{z^2}{c^2}\right)$).
Special case: for $a=b=c=R$, this reduces to the familiar sphere volume $\frac{4\pi R^3}3$.

Result

The volume is:

$$ V = \frac{4\pi abc}3. $$