Step 1 :

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

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For rotation about the y - axis, the surface area formula : $\"\"$ .

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The function is $\"\"$ .

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Solve for x.

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$\"\"$

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The function is $\"\"$ .

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Differentiate the curve with respect to y .

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$\"\"$

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Evaluating boundaries :

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

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

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Substitute corresponding values in $\"\"$ .

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The surface area is $\"\"$ .

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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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Substitute corresponding values in $\"\"$ .

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$\"\"$

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The area of the surface by rotating the curve about the y - axis is 52.391 square units.