Formulas of Arc Length

Some important results and formulas regarding the arc length of the curve are listed here.

1.  The arc length for a Cartesian curve $$y = f\left( x \right)$$ is given by \[S = \int\limits_{{x_1}}^{{x_2}} {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} } dx\]

2.  If the parametric form $$x = f\left( t \right)$$ and $$y = g\left( t \right)$$ is given, then \[S = \int\limits_\alpha ^\beta {\sqrt {{{\left( {\frac{{dy}}{{dt}}} \right)}^2} + {{\left( {\frac{{dv}}{{dt}}} \right)}^2}} } dt\,\,\,{\text{where}}\,\,\,\alpha \leqslant t \leqslant \beta \]

3.  The arc length of the equation of a curve in a polar system i.e., $$r = f\left( \theta \right)$$, is given by \[S = \int\limits_{{\theta _1}}^{{\theta _2}} {\sqrt {{r^2} + {{\left( {\frac{{dr}}{{d\theta }}} \right)}^2}} } d\theta \]

4.  If the equation of the curve is given by $$\theta = f\left( r \right)$$, then arc length is given by \[S = \int\limits_{{r_1}}^{{r_2}} {\sqrt {1 + {{\left( {r\frac{{d\theta }}{{dr}}} \right)}^2}} } dr\]