$\gamma(t)=Re^{it}$ for $0\le t \le 2\pi$, $R\in \mathbb{R}$.
In my notes it said the length of $\gamma$ =$\int_0^{2\pi}|\gamma'(t)|dt=2\pi R$.
Intuitively, it makes sense that it is the circumference of the circle, but calculating it out, I get a different answer?
$\int_0^{2\pi}Rie^{it}dt=Ri[\frac{e^{it}}{i}]_0^{2\pi}\\=R[e^{2\pi i}-e^0]\\=R[e^1-1]?$
The reason why I'm trying to calculate $\gamma$ =$\int_0^{2\pi}|\gamma'(t)|dt$ is because it is needed for the estimation lemma.