Earlier this week, a teacher posted in the College Board’s AP Calculus Community a request for an explanation of computing the arclength of a curve without relying on formulas.
The following video is my proposed answer to that question. In it, I derive the fundamental arclength relationship before computing the length of from x=0 to x=3 four different ways:
- As a function of x,
- As a function of y,
- Parametrically, and
- As a polar function.
In summary, the length of any differentiable curve can be thought of as
where a and b are the bounds of the curve, the square root is just the local linearity application of the Pythagorean Theorem, and the integral sums the infinitesimal roots over the length of the curve.
To determine the length of any differentiable curve, factor out the form of the differential that matches the independent variable of the curve’s definition.
OK, no CAS here, but the power of screencasting allows one to add a voice-over to problems that are difficult to capture in the linear text that predominates Web-based writing. The following two problems were requests from March 6-7, 2012 on the AP Calculus EDG. My solutions are screencasts using the ShowMe Whiteboard app on my iPad. Thanks to my former student (now technology consultant) Beth Holland for the lead to that great app.
- A region is defined by , , , and the x-axis. What is the volume of the solid that results when that region is revolved around the line ? Click here for my solution.
As an aside, colleague John Burk (@occam98) pointed me to a cool article, Volumes of Revolution & 3D Modeling, that quite rightly identifies the most challenging part about volumes of revolutions for students: visualization. This is another reason why we need to make greater use of screencasting or other technologies to get past the walls of words.
- A curve is defined parametrically by and . What is the length of that curve for ? Click here for my solution.
I find that many students (and some teachers) have difficulty sorting out the different forms of arclength. My screencast describes very briefly why I believe students should work from the single arclength formula, .