Volume by Cylindrical Shells

shells_1On Monday, June 15, I modeled a volume by cylindrical shells from Calculus II.  I used Example 1 in 7.3 of Stewart’s Essential Calculus, which is a volume of revolution of the curve \(y=2x^2-x^3\) about the y-axis. This is shaped a bit like a stadium. The plan is to approximate this volume using 16 cylindrical shells. I first sketched out the curve in 2-dimensions to get a feel for the profile of the shape. Next, I wrote out each point on the curve from \(x=0\) to \(x=2\) in intervals of length 1/8, namely \((0,0), … , (1.875,0.439), (2,0)\).

Each cylindrical shell is determined by its height, the thickness of the shell and either the inner or outer radius. By construction, each shell has thickness 1/8 and for each \(k=0,1,2, \dots, 15\), the inner radius of a shell is \(k/8\), while the outer radius is \((k+1)/8\). I chose the height of each shell to be \(f(k/8)\), which is the the \(x\)-value closer to the origin.

To summarize, if \(r=\)inner radius of shell, \(R=\)outer radius of shell, and \(h=\)height of tube, then for \(k = 0, 1, 2, \dots , 15\),

$$r = k/8,  \quad R = (k+1)/8,  \quad h=2r^2 – r^3.$$

Since the height function is increasing between \(x=0\) and \(x=4/3\), the cylindrical shells lie inside the volume of revolution. Between \(x=4/3\) and \(x=2\), the function is decreasing and the shells lie outside the volume.

shells_2I then made a model of this Riemann approximation for the solid in Cinema 4D by inserting tubes (Cinema 4D’s name for “cylindrical shells”) of inner radius \(r\), outer radius \(R\), and height \(h\). When Cinema 4D inserts a tube, it places half of the tube above the \(xy\)-plane (the \(xz\)-plane in Cinema 4D) and half of it below. Therefore, I needed to add half of each tube’s height to put each shell onto the \(xy\)-plane.

The object was first printed with the supports setting on just in case, although I thought that they did not need supports. The result was that a couple superfluous strands of plastic running along the shells, which needed to be removed. Given the geometry of the shape, I would advise against using supports in building this object.

I would advise that you use a raft to build this object, however, because it was very difficult to remove the object from the build plate. (It took 5 minutes of very careful tugging by David Pfaff.)

You can find this objects on Thingiverse here.