Prove that the change in length of a string is $\mathcal{O}[(\partial \psi / \partial x)^2].$

Question

For a string supporting a transverse wave, prove that $$ \int_0^{a}\sqrt{1 + (\partial \psi / \partial x')^2} \, dx' - a = \mathcal{O}[(\partial \psi / \partial x)^2]. $$ I have that $$ \int_0^{a}\sqrt{1 + (\partial \psi / \partial x')^2} \, dx' - a = \int_0^{a}\frac 12(\partial \psi / \partial x')^2 + \, ... \, dx' = \frac 12 a \big(\frac{\partial \psi(b)}{\partial x}\big)^2 +\, ... = \mathcal{O}[(\partial \psi / \partial x)^2], $$ where the mean value theorem for integration was used for the 3rd expression and $0<b<a$. Is this a sufficient argument? Is there a better way of showing this?