Problem :
What is the value of the integral $$ I = \int_{\gamma} \bar{z} dz,$$ where $\gamma$ is the lower part of the ellipse $$ x^{2}+9y^2=1 $$ running from (-1,0) until (1,0). So what I did, is a parametrization of the ellipse: $$ \gamma(t) = \cos(t) + \frac{i}{3} \sin(t).$$ And after I did the integration like this: $$ I =\int_{γ} \bar{z} dz = \int_\pi^{2\pi} \overline{\gamma(t)}\cdot\gamma'(t) dt,$$ and the result I got was $I = i \frac{\pi}{3}$ .
My question is, did I take the limits for the integral right? And is the method I used right?
( I changed the limits from $ [\pi,0]$ to $[\pi, 2\pi] $ thanks to ChristianBlatter at comments, so this question is answered)