# 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> | + | 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> | + | 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 | + | 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 01:05, 27 July 2019

To find the arclength of a curve , chop it up into infinitely small pieces of length as follows:

To find the arclength of a curve , chop it up into infinitely small pieces of length as follows:

Then add up all the lengths of the pieces to get the length of the whole curve: .

When you evaluate this integral, you get the arclength .

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 and convert to at the end.

To get the length of the section of curve between and , just add up the pieces between and :

To get the length of the section of curve between and , just add up the pieces between and :