How to calculate the index number for a curve around a linear system's fixed point without integrals?

Question

We know $$ \phi = \tan^{-1} \frac{\dot{y}}{\dot{x}},$$ yet so far I've only been able to calculate the index of curves by using the integral $$ \frac{1}{2\pi} \oint_C \frac{\dot{y}\ddot{x} - \dot{x}\ddot{y}}{\dot{x}^2 + \dot{y}^2} dt. $$ I'm only working with simple $2\times 2$ linear system fixed points, i.e. centers, saddles, stable nodes, etc... While my method works, with $x = \cos t$, $y = \sin t$, and $t \in [0,2\pi]$, I was told very quickly by my prof. that we can also directly calculate it with only the difference of $\phi$ at $t = 0, 2\pi$. Yet, every parameterization has cancelled out to zero when I calculate arctan: when the index is suppose to 1. It seems like my parameterization will always cancel out since $0 \equiv 2\pi \pmod{2\pi}$. What am I missing?