Example 1:Determine the surface area created by rotating the curve y = f(x) about the y axis a full revolution.

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Solution: Since the revolution is about the y axis, we only need to find the X-bar coordinate of the curve. We proceed by identifying the differential length along the curve as

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We then calculate X-bar as

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Notice that the integral in the denominator is the total length of the curve. Now, using the theorem of Pappus and Guldinus, we find the surface area as

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where equation in this case and equation. Thus, the surface area is found to be

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