**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
coordinate of the curve. We proceed by identifying the differential
length along the curve as

We then calculate 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 in this case and . Thus, the surface area is found to be