Compute the integral $\int_\gamma|z|^2\,\mathrm dz$ where $\gamma$ is the contour starting at $0$, going vertically to $i$, then going horizontally to $i + 1$, then going vertically to $1$, then, finally, going horizontally to $0$.
I think i need to parameterise the contours but I'm not sure how?
A natural way of parametrizing the first part of that contour (going from $0$ to $i$ vertically) is to define $\gamma(t)=it$ ($t\in[0,1]$). And$$\int_\gamma|z|^2\,\mathrm dz=\int_0^1|it|^2i\,\mathrm dt=i\int_0^1t^2\,\mathrm dt=\frac i3.$$Can you do the rest?