Understanding the Maths Behind Space Elevators

[RooftopDuvet]

I'm doing some research on space elevators and have found a site (http://www.zadar.net/space-elevator/#transverse") which gives some insight into the maths behind the elevator cable. If you click on the link and scroll down, up, whichever direction to equation (2) and a picture of a blue trapezium the guy has found the difference in the volume of the tapering cable with what looks like volume of revolution, but I can't seem to figure out what he's done. He says that it's 'easy to show...'. Am I missing something? I could use a little help. thanks.

 

[quantumdude]

Quick way to show this: You are dealing with finding the volume of a frustum of a cone. The volume $V$ of a cone is $V=\frac{1}{3}Ah$, where $A$ is the area of the base and $h$ is the height.

You can get the volume of the frustum illustrated in the figure by repeatedly applying the above formula and subtracting volumes, as follows:

$$V=V_{frustum+missing top}-V_{missing top}$$

Long way to show this: You can calculate the volume with a single integral because the frustum is a surface of revolution. Find an equation for the right boundary of the figure (it's a straight line segment, so that's easy) and solve it for x. Then revolve that line segment about the y-axis and write down an expression for the differential volume. Then integrate over y, and you should get the same expression.