Find the volume of the ellipsoid (x²)/(a²) + (y²)/(b²) +

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

We slice the ellipsoid into elliptical cross-sections, exactly mirroring the classical sphere-volume derivation.

Step 1: Cross-section at height $z$

\[\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1-\frac{z^2}{c^2}\]

an ellipse with semi-axes $a_z=a\sqrt{1-z^2/c^2}$, $b_z=b\sqrt{1-z^2/c^2}$, for $-c\le z\le c$.

Step 2: Area of the cross-section

\[A(z) = \pi a_zb_z = \pi ab\left(1-\frac{z^2}{c^2}\right)\]

Step 3: Integrate using symmetry

\[V = 2\pi ab\int_0^c\left(1-\frac{z^2}{c^2}\right)dz = 2\pi ab\left[z-\frac{z^3}{3c^2}\right]_0^c = 2\pi ab\cdot\frac{2c}{3} = \frac{4\pi abc}{3}\]

\[\boxed{V = \frac{4\pi abc}{3} \text{ cubic units}}\]

Step 4: Special case — the sphere

When $a=b=c=r$: $V=\dfrac{4\pi r^3}{3}$, the familiar sphere volume. ✓

Summary

\[\boxed{V = \frac{4\pi abc}{3} \text{ cubic units}}\]

Key insight: The ellipsoid volume formula generalizes the sphere formula by simply replacing the single repeated radius $r$ (in $r\cdot r\cdot r$) with the three distinct semi-axes $a$, $b$, $c$ — the same cross-sectional-slicing argument that derives the sphere's volume extends immediately to any ellipsoid.