$\"\"$

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Suppose the curve has the vector equation $\"\"$ , $\"\"$ , or, equivalently, the parametric equations $\"\"$ , where $\"\"$ are continuous.

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If the curve is traversed exactly once as t increases from a to b, then it can be shown that its length is $\"\"$ . $\"\"$

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The curve is $\"\"$ and the interval is $\"\"$ .

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Consider $\"\"$ .

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Differentiate the curve with respect to $\"\"$ .

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$\"\"$ .

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Consider $\"\"$ .

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Differentiate the curve with respect to $\"\"$ .

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$\"\"$ .

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Consider $\"\"$ .

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Differentiate the curve with respect to $\"\"$ .

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$\"\"$ .

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$\"\"$

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Length of the curve $\"\"$ over the interval $\"\"$ is

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$\"\"$

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$\"\"$

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

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Evaluating boundaries :

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At $\"\"$ , $\"\"$ .

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At $\"\"$ , $\"\"$ .

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Substitute corresponding values in $\"\"$ .

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$\"\"$

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Apply formula : $\"\"$ .

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$\"\"$

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$\"\"$

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Length of the curve $\"\"$ over the interval $\"\"$ is $\"\"$ .