$\displaystyle \int_{C}{z \ \ bar \ \ dz} $ , C is along the circle |z|=1

and

another case, C along the circle |z-1|=1.

$\displaystyle \int_{C}{\sqrt{z} dz} $. C is |z|=1, and |z-1|=1

i think the first part is the same for both cases which is 2pi*i.

and the second part is 0 for both cases by Cauchy Integral Theorem.

But i am not sure if that can be applied. Because sqrt z is only analytic on C\ { x axis } . Therefore, out circles are not connected?