I'm currently working through a step-by-step example which calculates the length of a circular arc $\gamma$ of radius $r$ subtended by angles $\theta_1$ and $\theta_2$.
The circular arc is parametrised as $\gamma (t) = c + re^{it}$ for $\theta_1 \leq t \leq \theta_2$.
To calculate the length, we need to work out the derivative, $\gamma'$, which is $\gamma'(t)=rie^{it}$.
Then by the definition of length, we get $$L(\gamma) = \int_{\theta_2}^{\theta_1}|rie^{it}| dt$$
I understand everything up until this point, but then the solution goes onto simplifying the integral further, and I do not understand why this can be done.
$$L(\gamma) = \int_{\theta_2}^{\theta_1}|rie^{it}| dt = r\int_{\theta_2}^{\theta_1}dt = r(\theta_2 - \theta_1)$$
Surely by integrating, you would get the following instead?
$$L(\gamma) = \int_{\theta_2}^{\theta_1}|rie^{it}| dt = r\int_{\theta_2}^{\theta_1}|ie^{it}| dt = r(e^{i\theta_2} - e^{i\theta_1})$$
Any help showing where I've gone wrong would be much appreciated!
$$ r\int_{\theta_{2}}^{\theta_{1}}|ie^{it}|dt \\ = r\int_{\theta_{2}}^{\theta_{1}}|i(\cos(t) + i\cdot \sin(t))|dt \\ = r\int_{\theta_{2}}^{\theta_{1}}|i\cos(t) - \sin(t)|dt $$
Now $i\cos(t) - \sin(t)$ parametrizes a unit circle on the real-imaginary axes. Remember that the absolute value of a complex number $|a + bi|$ is basically the magnitude as if it were a vector: $\sqrt{a^{2} + b^{2}}$. Since it's a unit circle, the magnitude will always be 1. That gives:
$$r\int_{\theta_{2}}^{\theta_{1}}1 dt = r(\theta_{2} - \theta_{1})$$