A note on arc length
Discrete Mathematics and its applications
Teaching Mathematics and Its Applications
Volume 28 Issue 3
Abstract
We consider how the arc length integral of the graph of a function in the plane is connected with the hyperbola and its rational parameterization.
Arc length calculations are frequently among the hardest problems I give my students in calculus I and II. Usually the difficulty stems from complicated integral forms whose solution is not immediately obvious and may require more sophisticated techniques that students have yet to master. Thus choosing the correct examples is of paramount importance. This article clearly identifies which elementary functions will provide nice, soluble integrals for arc length calculations.
Most of the integrals I do in calculus I are of type (A); in calculus II they are a mix of types (A) and (B), weighted more heavily toward type (A); and in calculus III I get the opportunity to introduce integrals of type (C). I am required to be exceedingly careful, even when selecting examples from the text, because it is so easy to get a form that is not integrable with the techniques of the class. This article, however, has given me the tools I need to create my own examples, as well as taught me what to look for in an example problem to guarantee a straightforward calculation.
The article will prove somewhat dense for readers whose background does not include much mathematics.
Also note that there is a typo in article. In the rational parameterization of the right arc of an hyperbola, the text lists two contradictory parameterizations for x. The second should actually say y=(1/2)(z-(1/z)).
what a useful article (as translated by you into how it applies to teaching and learning, of course)!
I liked it enough that I shared it with three colleagues and reproduced it for my entire calculus II class.