I'm doing a high school paper on surface area of ellipsoids. I can find the SA of a ellipsoid of revolution, but do not know how to find the SA for a scalene ellipsoid. I would like to avoid elliptic integrals, and instead somehow approximate the SA. What techniques are available to approximate multivariable functions?
Also, would it be easier to approximate the second equation $z(x,y)$ and then evaluate the double integral?
The goal is to approximate into a form such that i can evaluate the double integral in simple terms, without elliptic integrals.
There is an approximation proposed by Knud Thomsen in year $2004$ (have a look here) $$S\approx2\pi \Big((ab)^p+(bc)^p+(ac)^p \Big)^{\frac 1 p}\quad \text{where}\quad p=\frac{\log (3)}{\log (2)}$$
In the linked page, you will find discussions about the optimal value of $p$ as well as other formulae.
Using this online calculator for $a=2$, $b=3$, $c=4$ we get $S=111.604$ while the above formula gives $111.504$.
There is another approximation by the same author (Knud Thomsen) $$S\approx 4\pi \Big(\frac{(ab)^p+(bc)^p+(ac)^p}3 \Big)^{\frac 1 p}\quad \text{where}\quad p\sim 1.6075$$