Write the equation of the surface of revolution formed by revolving the profile curve

around the -axis.

To find the equation of the surface generated by revolving the curve around the -axis, we can use the formula for the surface of revolution. The general equation for the surface formed by revolving a function about the -axis is given by:

For our function , first, we need to compute the derivative:

Then, substituting and into the surface area formula:

The surface of revolution equation can thus be represented parametrically for any point on the surface as:

Where the angle varies from to to complete the revolution.

Now, think of it like a party hat. If you take that linear profile and spin it around, you’ll create a funky, fun surface that expands and narrows, just like our hat!