$\"\"$

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The parametric representation is $\"\"$ and $\"\"$ .

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Graph :

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(1).Draw the coordinate plane.

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(2).Graph the curve $\"\"$ and $\"\"$ .

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$\"\"$

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The length given by this formula is the length of the curve as it moves from $\"\"$ to $\"\"$ .

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The parametric representation is $\"\"$ and $\"\"$ .

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There is some number $\"\"$ , such that the curve traced between $\"\"$ will be same as the curve for $\"\"$ .

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Therefore, the length of the curve is $\"\"$ .

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

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The periodicity of $\"\"$ is $\"\"$ , the periodicity of $\"\"$ is $\"\"$ .

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The common periodicity of $\"\"$ and $\"\"$ is $\"\"$ , $\"\"$ .

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$\"\"$

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Period of $\"\"$ is $\"\"$ .

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To find the periodcity of $\"\"$ , find out the periodcity of $\"\"$ .

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Periodcity of $\"\"$ is $\"\"$ .

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Here $\"\"$ ,therefore periodcity of $\"\"$ is $\"\"$ .

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$\"\"$

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Periodcity of $\"\"$ is $\"\"$ .

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Periodcity of $\"\"$ is $\"\"$ .

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Therefore apply the integration from $\"\"$ to $\"\"$ .

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Consider $\"\"$

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$\"\"$ .

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Consider $\"\"$ .

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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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Where $\"\"$ .

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$\"\"$

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$\"\"$ .

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From the trignometric identity $\"\"$ .

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$\"\"$ .

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Use the graphing calculator to evaluate the above integral.

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By solving, the length obtained is $\"\"$ .

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$\"\"$

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

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Graph of the curve :

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$\"\"$ .