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TD 2/17/12:

This activity was done this year prior to any discussion of using $d\rr_1\times d\rr_2$ to do surface integrals, although this was hinted at during the previous lecture. One or two groups nonetheless (eventually) approached the problem this way, but most did not.

Most groups used the symmetry to chop in only one direction, as suggested by the given drawing. As usual, many students wanted to use $dz$ for the height of this ribbon. I tried to explain that, when unwrapping the ribbon and laying it flat on a table, the height is represented by lines perpendicular to the circles, and these lines were not vertical originally. This explanation did not appear to be helpful, although I do not believe this was due to an inability to see that these lines are tilted. Rather, I believe the problem lies in a lack of belief that the difference between the slant height and the vertical height matters.

An alternate explanation seemed to help, namely imagining the unfolded ribbon as a piece of tape. Sticking the tape on the cone clearly requires tilting the tape, so that it's final vertical height is less than it's width; you need more tape than the vertical height of the cone.

The combination of being able to ignore the fact that the ribbon is really a trapezoid, not a rectangle, but not being able to ignore the difference between slant height and vertical height is challenging for many students and some instructors. (One instructor tried to model a solution starting from the formula $A=\pi rs$ for the area of a cone!) This is the art of infinitesimal reasoning, or equivalently of making approximations, namely knowing when one can (and when one cannot) ignore small differences.

Infinitesimal reasoning is not however an art form, although it may feel that way at first. Rather, the rules are clear: Small quantities can only be ignored in comparison with something larger. For example, the difference between the area of a trapezoidal ribbon (the average circumference $2\pi(r+dr/2)$ times the height $ds$ of the ribbon) and the approximating rectangle ($2\pi r\times ds$ is clearly small (2nd order) compared to the area (1st order). But the slant height $ds$ is of the same order as the vertical height $dz$, so the difference between them matters.

One way to summarize such arguments is to count the number of “$d$”s appearing in each term of an expression, or equivalently to describe expressions with two $d$s as “doubly small”, etc. Be warned however that this particular example may be confusing, as it chops area in one direction only. More generally, expressions such as “$dA$” and “$dV$” may violate this counting rule, as they conventionally do not indicate how many ways they have been chopped. (A better notation might be to write $d^2A$ for area chopped in two ways, etc.) But these should be the only exceptions; we avoid for precisely this reason ever writing expressions such as “$dm$” for the mass of the cone, preferring to use the explicit combination $\sigma\,dA$.

A separate issue arose in my honors class, again with essentially no prior discussion of surface integrals. No groups in this class began by writing down what they were trying to find ($\int\sigma\,dA$) — and, with at most one exception, none of the groups used the slant height. Interestingly, several groups tried to check their answers by doing the computation two ways (using $r$ or $z$ as the parameter), and were puzzled as to why they obtained different answers.

I believe these students were not having difficulty seeing that the slant height mattered, but rather were writing down the only integrals they could imagine, namely ones in which the independent variables occurred in familiar combinations: $dx\,dy$, $r\,dr\,d\phi$, $r\,d\phi\,dz$, etc. Never having done a surface integral, these students did not recognize that such surface elements only work for surfaces with one coordinate constant.

This difficulty is surely related to students wanting to pull one coordinate out of a line integral, rather than using what they know to relate it to the chosen parameter. In both cases, a formulaic approach to doing integrals, rather than the idea of chopping and adding, is surely getting in the way.