The curve is  and the interval is .

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

.

Rewrite expression as .

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Differentiate the function with respect to .

Substitute corresponding values in .

The surface area :

Consider . \ \

Change of integral limits:

If then .

If then .

Substitute corresponding values in .

.

Therefore, the exact area of the surface by rotating the curve about the -axis is

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The exact area of the surface by rotating the curve about the -axis is

.