Show the length of a contour, given by traversing once round a circle radius r, is 2πr

Question

I have tried this problem using the definition for length of a contour $$ L(\gamma) = \int |\gamma'(t)| dt $$ Along the contour $\gamma =Z +re^{it}$

But I cannot get it to work out at $2\pi r$.

Answer 1

First we set: $$z(t) = r \cdot {e^{it}}\left\{ {\begin{array}{*{20}{c}} {\left\| {z(t)} \right\| = r} \\ {z'(t) = i \cdot z(t)} \end{array}} \right.$$ And with $$\gamma (t) = {z_0} + z(t)$$ we have $$\gamma '(t) = z'(t) = i \cdot z(t)$$ and $$\left\| {\gamma '(t)} \right\| = \left\| i \right\| \cdot \left\| {z(t)} \right\| = r$$ Ready to integrate: $$\int\limits_0^{2\pi } {\left\| {\gamma '(t)} \right\|dt = } \int\limits_0^{2\pi } {r \cdot dt} = r \cdot \int\limits_0^{2\pi } {dt} = 2\pi \cdot r$$