We have the parametric surface defined by
$x=\dfrac{1}{2}\big(1+b+(1-b)\color{red}{\cos(t)}\big)$
$y=\dfrac{-1}{2}(1-b)\sin(t)$
$z=2b$
with $b\in [1/2,1]$ and $t\in[-\pi,0]$.
Write $z=u(x,y)$ in order to calculate $u_x$ and $u_{xx}$.
Find the expression to calculate the surface integral of $u$ over its domain (it is not necessary to calcute its value).