Section Summary 9.2: Area of a Surface of Revolution

1. Definitions

   The surface area S of the surface obtained by rotating the curve y=f(x), with f positive and with a continuous derivative, , about the x axis is defined as

   

   Writing this formula using the notation given for arc length in section 9.1, we find that

   

   which we can think of as belts of rectangular area, of thickness ds and circumference .

2. Theorems

   None to speak of.

3. Properties/Tricks/Hints/Etc.

   Nothing special.

4. Summary

   We start with cones, or rather with the frustum of a cone. A surface revolved about an axis can be approximated as a bunch of ``frustra'', and as they get more and more numerous we might assume that the area of the approximation will converge to the area of the surface itself (the usual trick!).

   If you unwrap a cone of radius r and height (that is, with ``hypotenus side length'' l, and lay it flat, then we note a couple of things:

   1. arc length L is a linear function of , and
   2. so is area A.

   Solving for arc length L using the data points,

   

   we obtain .

   Solving for area A using the data points,

   

   we obtain .

   Hence, for the flattened cone,

   

   Now, for a frustum, you take a large cone's area and subtract off a smaller cone's frustum:

   

   or

   

   or

   

   which is equal to

   

   since

   

   by ``similar cones''!

   If we define as the average radius, then we can write

   

   Now

   

   so

   

   which becomes

   

   in the limit, where y>0 and ds is the arc length.