# Complex integral depending on the chosen path?

Let's say $$f:\mathbb C\backslash\{0\}\to\mathbb C$$ is holomorphic and $$\text{Res}(f,0)=1$$. Now if I look for example at the integral $$\oint_{|z|=2}f(z)dz$$ I get confused by the following: The path could for example be $$\gamma(t)=2e^{it}$$ where $$t\in[0,2\pi]$$, then the winding number of $$\gamma$$ around $$z=0$$ is exactly $$1$$ and the integral has the value $$2\pi i$$ according to the Residue Theorem. But can't we also use $$\gamma(t)=2e^{-it}$$ where $$\gamma$$ goes around $$z=0$$ clockwise and thus the winding number is $$-1$$? This would mean that the integral can also be $$-2\pi i$$. Where is my error?