$\"\"$

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

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

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$\"\"$ .

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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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$\"\"$

\

\

consider $\"\"$ .

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$\"\"$

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Change of integral limits:

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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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$\"\"$

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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 $\"\"$ .