Last year’s posts about Arc Length On The Cardiod, Vector Products, and integral-secant-cubed are relevant again. The cardiod was a recent in-class example; the “vector” note corrects a mistake I didn’t make until yesterday; the “surprisingly useful integral” is one I had students to look up in a table on the recent exam (and work with… *nobody* had this one all the way right but me though in principle it’s an “easy” problem when the table is available…).

What the heck. Here’s the problem: *The polar equation , for , determines an “Archimedean Spiral”, S. Determine the arc length along S for between 0 and (exactly; a messy “formula”… check the work numerically).* And the answer: . This evaluates to about .5466… as does … so we can be reasonably sure this is right.

In my own work messy expressions like this are almost never right the first time… on the copies of the handwritten solution I distributed yesterday, quite a bit of erased work is clearly visible. Rooting out every last little mistake until things are just right is of course a vital part of the process for a *lot* of problems. One student was *real* close.

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