Evaluate $\int_Tz\,\mathrm dz$ and $\int_T\overline z\,\mathrm dz$ where $T$ is the triangle with vertices $0,1,-i$ oriented clockwise.
I am trying to solve this question, but I'm unsure how to parameterize it. Can you explain how I get this to an integrable form?
From $0$ to $1:$ $$\gamma(t)=t, t\in [0,1].$$
From $1$ to $-i:$ $$\gamma(t)=1-t-it, t\in [0,1].$$
From $-i$ to $0:$ $$\gamma(t)=(t-1)i, t\in [0,1].$$
Thus,
$$\int_T zdz=\int_0^1 tdt+\int_0^1 (1-t-it)(-1-i)dt+\int_0^1(t-1)iidt\\=\left[\frac{t^2}{2}\right]_0^1-(1+i)\left[t-\frac{t^2}{2}-i\frac{t^2}{2}\right]_0^1 -\left[\frac{t^2}{2}-t\right]_0^1 \\=\frac12-(1+i)\left(\frac12-i\frac12\right)+\frac12=0.$$