## Section 4.3 | Curve Lengths and Surface Areas of Parametric Curves

Surprise! Our old friend \(ds\) is back. Recall from our derivation of the formula for \(ds\), we reached a point where we said \[|\overline{P_{i - 1}P_i|} = \sqrt{(\Delta x_i)^2 + (\Delta y_i)^2}.\]We just need a slight adjustment from here to get to a parametric form of this curve length. We know from the Mean Value Theorem for Derivatives that there is some point \(t_i^* \in [t_{i - 1}, t_i]\) and some \(t_i^{**} \in [t_{i - 1}, t_i]\) such that \[\Delta x_i = f(t_i) - f(t_{i - 1}) = f'(t_i^*)(t_i - t_{i - 1}) = f'(t_i^*) \Delta t\]\[\Delta y_i = g(t_i) - g(t_{i - 1}) = g'(t_i^{**})(t_i - t_{i - 1}) = g'(t_i^{**}) \Delta t\]Hence, \[|\overline{P_{i - 1}P_i}| = \sqrt{(\Delta x_i)^2 + (\Delta y_i)^2} = \sqrt{\left[f'(t_i^*) \Delta t\right]^2 + \left[g'(t_i^{**}) \Delta t\right]^2} = \sqrt{\left[f'(t_i^*)\right]^2 + \left[g'(t_i^{**})\right]^2} \Delta t\]