As the title says, how do we integrate $|z|dz$ on a straight line on the complex plane? Suppose that I've already known the parametrization. If it were on reals, we would break the integral down to multiple parts where z changes sign, but how do I do that on the complex plane?
Is it just simply the formula for absolute value (square root of the sum of the squares of real and imaginary parts)?
HINT:
Along the straight-line path from $z_1$ to $z_2$ we have
$$\begin{align} \int_{z_1}^{z_2}|z|\,dz&=\int_0^1 |z_1+(z_2-z_1)t|\,(z_2-z_1)\,dt\\\\ &=(z_2-z_1)\int_0^1 \sqrt{|z_1|^2+|z_2-z_1|^2t^2+2\text{Re}(z_1^*(z_2-z_1))t}\,dt \end{align}$$