I am looking for direct proofs that the total curvature $\int_0^{L_\gamma} \! \gamma''(s) \, \mathrm{d} s$ of any Jordan curve $\gamma$ resp. $\int_0^{L_\gamma} \! |\gamma''(s)| \, \mathrm{d} s$ of a convex Jordan curve equals $2\pi$. Direct proof means: a calculation and not a special case of the theorem of Gauss-Bonnet or of 
I have no idea even how to show that the integral
$$\int_0^1 \frac{\mathrm{d} x}{(1-(1-b^2)x^2)^{3/2}}$$
evaluates to $\frac{1}{b}$ which would prove the above at least for ellipses with major axis $a=1$ and minor axis $b$. Let alone for arbitrary (convex) curves/functions.
