Evaluate $$\int_{\gamma} (z^2-2) \mathrm{d}z$$ where $\gamma$ is the following curve:
Use two methods: direct calculation via a parametrization of $\gamma$, and the fundamental theorem.
I know about the fundamental theorem, so I simply evaluate $\frac{z^3}{3}-2z$ at $3$ and $0$, the end points of the curve. My problem is that I don't know how to parametrize spirals such as this for the second method.
The idea is to let the radius vary with the angle, so try something like $\gamma(t) = cte^{it}$, and let's find $c$. We have $\gamma(0) = 0$, and I want $\gamma(6\pi) = 3,$ so: $$6c\pi e^{6i\pi} = 3 \implies 6c\pi = 3 \implies c = \frac{1}{2\pi}.$$
I took $6 \pi$ because of the drawing. We are going counterclockwise and we went around three times. We could use the same line of thought for $c_1te^{ic_2t}$, but I didn't see how changing the speed could help us. Try $\gamma(t) = \frac{t}{2\pi}e^{it}$, then.