I need help, I'm studying the area of surfaces using double integrals, I attempted to solve this exercise but I'm having trouble. The problem is to compute the area of that part of the surface of the sphere $x^2 + y^2 + z^2 = a^2$ cut out by the surface $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$. This exercise is from the book "Problems-in-Mathematical-Analysis-Demidovich".
I know that the equation of the sphere is $z = \sqrt{a^2-x^2-y^2}$ , and $\frac{dz}{dx} = -\frac{x}{\sqrt{a^2-x^2-y^2)}}$ , and $\frac{dz}{dy} = -\frac{y}{\sqrt{a^2-x^2-y^2}}$. Then I have to use the formula $S = \sqrt{1+\left(\frac{dz}{dx}\right)^2 + \left(\frac{dz}{dy}\right)^2}$. Here I'm having trouble, I don't know why the solution guide multiplies the formula $S$ by $8$.
Can the somebody explain me why is this, please.