$\"\"$

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The parametric equations that represent the ellicpse are $\"\"$ and $\"\"$ .

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The initial condition is $\"\"$ .

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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 value of $\"\"$ .

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$\"\"$

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From the trignometric identity : $\"\"$ .

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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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As we considered the first quadrant, which is from $\"\"$ to $\"\"$

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Therefore $\"\"$ and $\"\"$ .

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Use symmetry change the limits to first quadrant only, multiplying by $\"\"$ .

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$\"\"$

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Apply Integration :

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$\"\"$

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$\"\"$

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The total length of the astroid is $\"\"$ .