Let $\mathbb \Gamma$ denote the boundary of a convex polygon with vertices $w_1, ..., w_n$ in $\mathbb C$.
Show that
$$\int_{\mathbb \Gamma}\lambda^kd\lambda = 0 , \forall k \ge 0$$
I've found some information regarding the question which could help me approach this (in the picture below). But I'm still struggling how to use contour integration in order to calculate the integral (show that it's equal to zero), when $f(\lambda) = \lambda^k$ for $k \ge 0$.
And also, another approach I was thinking of, using the following Theorem in order to solve the question:
Some helpful background:
