Arclength

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To find the arclength of a curve $y=f(x)$, chop it up into infinitely small pieces of length $ds$ as follows:

CartesianArclength



To find the arclength of a curve $r=f(\theta)$, chop it up into infinitely small pieces of length $ds$ as follows:

CartesianArclength



Then add up all the lengths of the pieces to get the length of the whole curve: $\int ds$.

When you evaluate this integral, you get the arclength $s$.



It is clear from the diagrams that the polar coordinate version is way harder than the cartesian coordinate version, so just work in terms of $x,y$ and convert to $r,\theta$ at the end.



To get the length of the section of curve between $x=a$ and $x=b$, just add up the pieces between $x=a$ and $x=b$: \[\int_{x=a}^b ds=\int_a^b ... dx.\]



To get the length of the section of curve between $\theta=a$ and $\theta=b$, just add up the pieces between $\theta=a$ and $\theta=b$: \[\int_{\theta=a}^b ds=\int_a^b ... d\theta.\]

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