What is the surface area of the 3D saddle surface $f$ defined by $$ f = [x, y, z(x,y)] $$ and with $$ z(x,y) = \frac{k_1}{2} x^2 + \frac{k_2}{2} y^2 $$ for $(x,y)\in(-1,1)\times(-1,1)$ and for any $k_1,k_2\in R$?
I am trying to compute it analytically using $$ A = \iint_{x,y\in(-1,1)^2} \sqrt{\det(I(x,y))} dx dy = \iint_{x,y\in(-1,1)^2} \sqrt{1 + k_1^2 x^2 + k_2^2 y^2} dx dy $$ where $I$ is the first fundamental form of $f$.
Does any analytical solution exist for this integral?
Additional questions: What is the surface area of $f$ for
- $(x,y)$ such that $x^2 + y^2 < 1$ (integral over the unit disk)
- $(x,y)$ such that $x^2 + y^2 + z(x,y)^2 < 1$ (integral over the intersection of the surface with the unit ball)