Arc Length

For a smooth curve given by $y=f(x),$ the arc length of $f$ between $a$ and $b$ is

$$s=\int_a^b \sqrt{1+[f'(x)]^2}\;dx.$$

For a smooth curve given by $x=g(y),$ the arc length of $g$ between $c$ and $d$ is

$$s=\int_c^d \sqrt{1+[g'(y)]^2}\;dy.$$