I know that there is a mistake here, but I need help finding what my error actually is! I know that surface area of a sphere of radius r is $\displaystyle 4\pi r^2$. However from the figure:

radius of a cross-section: $\displaystyle x=\sqrt{r^2-z^2}$

Circumference of a cross-section: $\displaystyle C=2\pi \sqrt{r^2-z^2}$

So, one should be able to integrate to get the surface-area, but the answer does not agree:

$\displaystyle SA=2\pi \int _{-r}^r \sqrt{r^2-z^2}dz=\pi ^2r^2$