Parameterizing to evaluate a line integral with complex numbers

Question

I'm trying to evaluate $\int_C(z^2+3z)dz$ along the circle $|z|=2$ from (2, 0) to (0, 2) going counterclockwise. I have an answer, but I was told it was wrong. It apparently should be $\frac{-44}{3}-\frac{8}{3}i$. Here's what I did: $$z=e^{i\theta}, dz=ie^{i\theta}, 0\leq\theta\leq\frac{\pi}{2}$$ $$\int_{0}^{\frac{\pi}{2}}\left(e^{2i\theta}+3e^{i\theta}\right)ie^{i\theta}=i\int_0^{\frac{\pi}{2}}\left(e^{3i\theta}+3e^{2i\theta}\right)=i\left[e^{3i\theta}+3e^{2i\theta}\right]_0^\frac{\pi}{2}=\frac{-10}{3}-\frac{i}{3}$$

Where am I going wrong?

Answer 1

You should have $z=2e^{i\theta } $ because of $|z|=2$ to start with . Try again and see what happens.

Answer 2

Note that you are integrating over an arc of a circle of radius 2. Therefore, your paramaterisation should be $z=2e^{i\theta}$ with $0\le \theta\le\pi/2$. Also, it is appropriate to write $dz=2ie^{i\theta}d\theta$ and to write $d\theta$ in your integrals.