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


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

equation equation

We then calculate X-bar as


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


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