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 yaxis. Any help you can provide would be greatly appreciated. Jason 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 xaxis. In the diagram below then you want to revolve either the region shaded red or the region shaded blue around the yaxis. 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 yaxis. This is the region bounded by y = 0 x = 0 and x = b
Take a strip of thickness deltax, at r on the xaxis, as in the diagram and rrevolve it about the yaxis. This gives a cylindrical shell of radius r, height h and thickness deltax, and hence volume
The volume required is thus
You can evaluate this integral using integration by parts with dv = cosh(x/a) dx Harley
