Complex Line Integral of absolute value of z

Question

How do we proceed for the following complex line integral?

$$\int\limits_\gamma |z|\:dz$$

where $\gamma$ is the half circular $|z|=1$, $0\leq \arg (z) \leq \pi$

taking $z=1$ as the initial point.

Any hints would be helpful. Thanks.

Answer 1

Substitute $z = e^{i\theta}$. The limits become $0$ to $\pi$

$|z| = 1$ and $dz$ = $ie^{i\theta}\:d\theta$

$$\int\limits_\gamma |z| \:dz = \int\limits_0^\pi ie^{i\theta} \:d\theta$$