Surface of $(x^2 + y^2 + z^2)^2 = a^2 * (x^2 - y^2)$ using surface integrals

Question

I have to find the surface of $$(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2)$$ using a surface integral and really have no idea what to do... I would really appreciate it if you could give me an idea.

Answer 1

To find $S$ may be highly non-trivial: that implicit surface recalls some equipotential surfaces; a similar (just 2d) question of mine is still unsolved.

Anyway, the section $z=0$ of that surface is a lemniscate, and by setting: $$ f(x,y) = \sqrt{-(x^2+y^2)+|a|\sqrt{x^2-y^2}} $$ we have: $$ S = 8\iint_{L}\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1}\;dx\,dy$$ where $L$ is the region given by $0\leq x\leq a$ and: $$ 0\leq y\leq \frac{1}{\sqrt{2}}\sqrt{-a^2-2x^2+|a|\sqrt{a^2+8x^2}}. $$