Analyzing a Complex Integral: Circle of Radius 2 Centered at 0 Counterclockwise

[player1_1_1]

Homework Statement

integral: $$\int\limits_C\frac{\mbox{d}z}{z}$$ where $$C$$ is circle of radius 2 centered at 0 oriented counterclockwise

Homework Equations

The Attempt at a Solution

I am going to parameter this: $$\gamma=2\cos t+2i\sin t,\ \gamma^\prime=-2\sin t+2i\cos t,\ t\in[0,2\pi]$$, then $$z=x+iy=2\cos t+2i\sin t$$ and integral will look like this:
$$\int\limits^{2\pi}_0\frac{-2\sin t+2i\cos t}{2\cos t+2i\sin t}\mbox{d}t=\int\limits^{2\pi}_0\frac{i\left(i\cos t-\sin t\right)}{i\cos t-\sin t}\mbox{d}t=2\pi i$$
is it correct? and another question, what is counterclockwise? thanks for answer

 

[HallsofIvy]

"Counter Clockwise" (called by our British cousins "anti-clockwise") means going around a circle opposite to the way clock hands do. On a coordinate system, going around the circle with radius 2 from (2, 0) to (0, 2) to (-2, 0) to (0, -2) back to (2, 0).

Yes, what you have done is correct. You could also use the parameterization $z= 2e^{i\theta}$.

 

[player1_1_1]

oh, sure, thanks for answer;)