Subject: Volume

My name: Jason

Difficulty level: College

I am a high school math teacher. I was asked by a friend who is in architectural design for a method for determining the volume of what he called a Catenary.

The Catenary curve is modeled by the equation y=a cosh(x/a). I ran into a mess when I tried to compute the volume of the solid formed by revolving that curve around the y-axis.

Any help you can provide would be greatly appreciated.


Hi Jason,

The catenary is a very interesting curve. It is the shape taken by a flexible cable (like a power line) suspended between two towers of the same height. The name comes from the Latin word for chain.

I am going to assume that the piece of the catenary that you want stetches form -b to b on the x-axis. In the diagram below then you want to revolve either the region shaded red or the region shaded blue around the y-axis. If you can find one volume you can find the other since their sum is the volume of a right, circular cylinder.

I am going to find the volume obtained by revolving the region shaded blue around the y-axis. This is the region bounded by

y = a cosh(x/a)
y = 0
x = 0 and
x = b
I am going to use the method of cylindrical shells.

Take a strip of thickness delta-x, at r on the x-axis, as in the diagram and rrevolve it about the y-axis.

This gives a cylindrical shell of radius r, height h and thickness delta-x, and hence volume

The volume required is thus

You can evaluate this integral using integration by parts with

u = x

dv = cosh(x/a) dx

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