$\"\"$

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The parametric equations of the curve are $\"\"$ and $\"\"$ .

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The interval is $\"\"$ .

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Since the particle retraces its route the limit of integration for the length is from $\"\"$ to $\"\"$ .

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Consider $\"\"$ .

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Diffrentiate with respective to $\"\"$ .

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$\"\"$

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Consider $\"\"$ .

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Diffrentiate with respective to $\"\"$ .

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$\"\"$

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$\"\"$

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Find the distance travelled by the curve.

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$\"\"$ .

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Where $\"\"$ and $\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$

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Consider $\"\"$ .

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By applying simmentry :

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$\"\"$ .

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Let $\"\"$ , $\"\"$ .

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Substiute $\"\"$ .

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Substiute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Find the length of the curve.

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Theorem :

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If a curve is described by the parametric equations $\"\"$ and $\"\"$ , $\"\"$

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then the length of the curve is $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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The length of the curve is $\"\"$ .

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The distance of the curve is $\"\"$ .