How would I evaluate $\int \bar z dz$, with
1: the contour $\gamma$ being the straight line segment from $0$ to $1+i$
2: the contour $\sigma$ being the straight line segment from $0$ to $1$, followed by the straight line segment from $1$ to $1+i$.
For the first one, I tried working it from the definition $\int_\gamma f(z) dz =\int_a^bf(\gamma(t))\gamma'(t) dt$. The only question I have is, is defining $\gamma(t)=t+it$, $t\in [0,1]$ correct?
For the second one, how would I define the contour $\sigma$ and thus evaluate $\int \bar z dz$? I can't seem to be able to define $\sigma$.
Alternatively, is there any other way to evaluate $\int \bar z dz$ apart from going straight from the definition?