Evaluating $\int_\gamma |z|\,|dz|$ I'm updating my question to reflect changes that have occurred.
I'm kind of stuck trying to figure out this integral:
$$\int_\gamma |z|\,|dz|=\int_{0}^{\pi}|t||1+t||dt|$$
where $\gamma(t)=te^{it}$ where $0 \leq t \leq \pi$.
Since I have reduced this to $t$, do I approach this for $-t$ and $t$?

I understand the question as asking for
$$I = \int_\gamma |z| |dz|$$
where the curve $\gamma$ is defined to be the locus of $z(t)$ from $t=0$ to $t=\pi$ with $z(t) = t e^{it}$.
Then $|z| = t$ and $|dz| = |e^{it} + ite^{it}| = |1+it| = \sqrt{1+t^2}$, so
$$\begin{eqnarray}
I & = & \int_0^\pi t\sqrt{1+t^2} dt \\
  & = & \frac12 \int_{t=0}^{t=\pi} \sqrt{1+t^2}d(1+t^2) \\
  & = & \frac13 \left[(1+t^2)^\frac32\right]_{t=0}^{t=\pi} \\
  & = & \frac{(1+\pi^2)^\frac32 - 1}{3} \\
  & = & 11.612\dots
\end{eqnarray}$$
