Evaluate $\int \bar z dz$ 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?

Your $\gamma$ for 1) is fine. For 2), just choose a path which connects $0$ and $1$ on, say, $[0,1]$ and $1$ and $1+i$ on $[1,2]$ such that path is continuous and piecewise differentiable on $[0,2]$. 
You could, for example, define $\sigma(t) := t$  (which is the same as $ t +0i$) for $0 \le t \le 1$ and $\sigma(t) := 1+ (t-1)i$ for $1 \le t \le 2$
