Find with proof the volume of the region that is contained within $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ and with $\displaystyle a,b,c \neq 0$.

Sure I know this can be done with calculus (multiple integrals or volume from rotation) but if that were the kind of solution I was looking for I would have posted in the Calculus section.

Hint: Orthogonal Projections