Difference between $\frac{x^2} {a^2}+\frac{y^2}{b^2} +\frac{z^2} {c^2} = 1$ and $\frac{x^2}{a^2} + \frac{y^2}{b^2} +\frac{z^2}{c^2}\leq 1$?

Question

Is the volume of the ellipsoid $\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} +\dfrac {z^2} {c^2} = 1$ the same as the volume of $\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} +\dfrac {z^2} {c^2} \leq 1$?

Answer 1

Strictly speaking, the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ defines the boundary of $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\leq 1$. The volume of the first equation is 0. You could think of the first equation as an empty ellipsoid shell and the second equation as a filled in ellipsoid.

This is similar to the unit circle being defined by $x^2+y^2=1$, but the filled in unit circle being defined by $x^2+y^2\leq 1$.