Solution
= Solution
The arclength element is $ds=\sqrt{1+y'^2+z'^2}\,dx$. For uniformly small slopes, the <Taylor expansion> $\sqrt{1+s}=1+s/2+O(s^2)$ gives
$$
\ell=L+\frac12\int_0^L(y'^2+z'^2)\,dx
+O\left(\int_0^L(y'^2+z'^2)^2\,dx\right).
$$
Thus \b[to quadratic order in the slopes] the excess length is $\boxed{\ell-L\simeq\tfrac12\int_0^L(y'^2+z'^2)\,dx}$. This is the small-slope approximation used in the subsequent <elastic energy>.