Parametrizing Complex Line Integral

[Newtime]

So this is an ultra basic question, but I'm rusty with parametrization techniques and wanted to make sure I was doing this correctly. Let's say I want to evaluate $$\int_{\gamma} z \: dz$$ where $$\gamma : [a,b]\rightarrow \mathbb{C}$$ is some path of integration. Now, I figure I can parametrize the curve and apply the definition of complex integration to arrive at the following: $$\gamma(t) = x(t) + iy(t) \quad \text{so} \quad \int_{\gamma} z \: dz = \int_a^b \gamma(t) \gamma(t)' \: dt = \int_a^b (x(t)+iy(t))(x'(t)+iy'(t)) \: dt$$ and distribute from there. Again, I know this is a very basic question, and I'm pretty sure it's correct, but it's been a while so I wanted to make sure I wasn't making some silly logical error (quite possible). Thanks.

 

[arkajad]

So far so good. You are following the definition. You can also calculate this way

$$\int_\gamma f(z)dz$$

replacing $$f(z)$$ by $$f(\gamma(t))$$

 

[jackmell]

Shouldn't that be from t_0 to t_1?