Find the Area of the upper Cap cut from the sphere $x^2+y^2+z^2=2$ by the cylinder $x^2+y^2=1$.
I got how to solve it after seeing solution using $dS=\iint\sqrt{\frac{dz}{dx}^2 +\frac{dz}{dy}^2-1 } dxdy$
But in my attempt using spherical coordinates.. I got a different answer (which is wrong).. I don't understand what is the mistake. Please help me
$$\iint r^2\sin\theta d\theta d\phi$$ with $r^2=2$ and integrating with $\theta\in[\frac{\pi}{4},\frac{3\pi}{4}]$ and $\phi \in[0,2\pi]$
I was getting $4\sqrt{2}\pi$
There is an error in your specification of the $\theta$ range. The correct spherical integral for the upper cap cut should be
$$r^2\int_0^{2\pi} d\phi \int_0^{\pi/4}\sin\theta d\theta = 2\pi \left( 2-\sqrt{2} \right)$$