$\displaystyle $$\begin{eqnarray*} L & = & \int_{\alpha}^{\beta} \sqrt{ \left\{ f'(t) \right\}^2 + \left\{ g'(t) \right\}^2 }~dt\\[0.5em] & = & \int_{\alpha}^{\beta} \sqrt{ \left( \dfrac{dx}{dt} \right)^2 + \left( \dfrac{dy}{dt} \right)^2 }~dt \end{eqnarray*}$$ $