The shell model comes apart into its individual shells:

Thingiverse link: http://www.thingiverse.com/thing:305282

Settings: Printed on a Replicator 2 with .3mm/low default settings.

Technical notes, math flavor: (Apologies for the bad math typesetting is what is to follow; I haven't had luck with WordPress and math formatting or embedding LaTeX code.) These heights of these shells were determined using the midpoints of eight subintervals from

*x=*0 to*x=*2. The reason we use the midpoints is that it allows us to use a very nice formula for computing the volumes of the shells that will allow us to construct a Riemann Sum, and thus a definite integral, to express the exact volume. Suppose we've subdivided [0,2] into eight subintervals of the form [*x_{k-1},x_k*]. Then each shell is just a really tall washer with inner radius*x_{k-1}*, outer radius*x_k*, and height given by the function curve somehow. Let's take the height at the midpoint*m_k*= (*x_{k-1}*+*x_k*)/2 of the subinterval. Then our*k*th shell has volume*V_k*= pi * (

*x_k*)^2 *

*f*(

*m_k*) - pi * (

*x_{k-1}*)^2 *

*f*(

*m_k*).

By factoring out like terms we can turn this into:

Now using the fact that *V_k*= pi *

*f*(

*m_k*) [ (

*x_k*)^2 -(

*x_{k-1}*)^2].

*a*^2-

*b*^2 = (

*a*+

*b*)(

*a*-

*b*), together with the definition of

*m_k*and

*DeltaX*=

*x_k*-

*x_{k-1}*and a bit of algebra, we can write this volume as:

*V_k*= 2 * pi *

*m_k**

*f*(

*m_k*) *

*DeltaX*.

This formula isn't any easier to use in practice, say if we were to actually calculate and add up the volumes of the eight shells pictured above; however the form of this

*k*th volume expression - and in particular the presence of the

*DeltaX*in the expression - allows us to construct a definite integral that represents the

*exact*volume of the solid of revolution.

Technical notes, syllabus flavor: Calculus profs will notice that late in the semester is an odd time to talk about volumes with shells; if you're teaching Calc 2 then that's usually in mid-semester. Here at JMU we teach a two-semester Calculus I with Integrated Precalculus course that goes about a third of the way into Calc 2, so we get through Riemann sums, the Fundamental Theorem of Calculus, techniques of integration, and volumes/applications by the end of the second semester.

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