I have a general ellipsoid like $$\frac{x^2}{a^2} + \frac{y^2}{b^2} +\frac{z^2}{c^2}=1$$ which is cut at top and bottom as say $zTopCut$ and $zBottomCut$. How can I determine surface area on the curved surface of the ellipsoid? I need to implement in coding.
My idea was to divide $z$-axis into several ellipse shells. Then determine $a$ and $b$ for the mentioned $z$. Then find the surface area as $$4 \int_0^a {\sqrt{\left({ 1 +\frac{dy}{dx}}\right)}^2}dx.$$ But the result is not OK. I am confused.
Any help is appreciated. Thanks and Regards.