 Solution

The arclength element is $ds=\sqrt{1+y^{\prime 2}+z^{\prime 2}}dx$. For uniformly small slopes, the Taylor expansion $\sqrt{1+s}=1+s/2+O(s^2)$ givesThus to quadratic order in the slopes the excess length is $\boxed{\ell -L\simeq \frac{1}{2}{\int }_0^L(y^{\prime 2}+z^{\prime 2})dx}$. This is the small-slope approximation used in the subsequent elastic energy.