$\"\"$

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The surface area formula, for rotation about the $\"\"$ - axis is $\"\"$ .

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

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

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$\"\"$

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Substitute corresponding values in $\"\"$ .

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The surface area :

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$\"\"$

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

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Evaluating boundaries :

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If $\"\"$ , then $\"\"$ .

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If $\"\"$ , then $\"\"$ .

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$\"\"$

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Substitute corresponding values in $\"\"$ .

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$\"\"$

\

$\"\"$

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Therefore, the exact area of the surface by rotating the curve about the $\"\"$ - axis is $\"\"$ .

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$\"\"$

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The exact area of the surface by rotating the curve $\"\"$ about the $\"\"$ - axis is $\"\"$ .