Volume by Cylindrical Shells

On 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.

I 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.