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Difference between revisions of "Arclength"

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To find the arclength of a curve <math>y=f(x)</math>, chop it up into pieces of length <math>ds</math> then add them up to get <math>\int ds</math>, as follows:
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To find the arclength of a curve <math>y=f(x)</math>, chop it up into infinitely small pieces of length <math>ds</math> as follows:
 
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To find the arclength of a curve <math>r=f(\theta)</math>, chop it up into pieces of length <math>ds</math> then add them up to get <math>\int ds</math>, as follows:
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To find the arclength of a curve <math>r=f(\theta)</math>, chop it up into infinitely small pieces of length <math>ds</math> as follows:
 
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It is clear from the diagrams that the polar coordinate version is harder than the cartesian coordinate version, so instead work in terms of <math>x,y</math> and convert to <math>r,\theta</math> as needed.
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Then add up all the lengths of the pieces to get the length of the whole curve: <math>\int ds</math>.
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When you evaluate this integral, you get the arclength <math>s</math>.
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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 <math>x,y</math> and convert to <math>r,\theta</math> at the end.
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To get the length of the section of curve between <math>x=a</math> and <math>x=b</math>, just add up the pieces between <math>x=a</math> and <math>x=b</math>: <cmath>\int_{x=a}^b ds=\int_a^b ... dx.</cmath>
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To get the length of the section of curve between <math>\theta=a</math> and <math>\theta=b</math>, just add up the pieces between <math>\theta=a</math> and <math>\theta=b</math>: <cmath>\int_{\theta=a}^b ds=\int_a^b ... d\theta.</cmath>

Latest revision as of 02:05, 27 July 2019

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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