I'm working on the following problem:
Evaluate $\int_{\gamma} \overline{z}^{2}dz$ along $\gamma$, where $\gamma$ is the straight-line joining $(0,0)$ to $(1,1)$.
I defined $\gamma : [0,1] \rightarrow \mathbb{C}$ by $\gamma (t) = t+it$. Then using the definition of the contour integral,
$\int_{0}^{1} -2it^{2}(1+i)dt = \frac{2}{3}-\frac{2i}{3}$.
Is this correct? Any advice is appreciated!