Tag Archives: integral

A Generic Approach to Arclength in Calculus

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 y=x^2 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.

Calculus Humor

Completely frivolous post.

OK, my Halloween “costume” at school this year was pretty lame, but I actually did put a minute amount of thought into it.


In case you can’t read the sign, it says \int 3(ice)^2d(ice).  If you remember some calculus and treat ice as your variable, that works out to ice^3–An Ice Cube!  Ha!

But it gets better.  As there weren’t any bounds, adding the random constant of integration makes it ice^3+C or “Ice Cube + C”, or maybe “Ice Cube + Sea”–I was really dressed up as an Iceberg.  Ha! Ha!  Having no idea how to dress like an iceberg, I wore a light blue shirt for the part of the iceberg above the water and dark blue pants for the part below the water.  I tried to be clever even if the underlying joke was just “punny”.

Then a colleague posted another integral joke I’d seen sometime before.  It has some lovely extensions, so I’ll share that, too.

What is \int \frac{d(cabin)}{cabin}?

At first glance, it’s a “log cabin.”  Funny.

But notationally, the result is actually ln(cabin), so the environmentalists out there will appreciate that the answer is really a “natural log cabin.”  Even funnier.

The most correct solution is ln(cabin)+C.  If you call the end “+ Sea”, then the most clever answer is that \int \frac{d(cabin)}{cabin} is a “Houseboat”.  Ha!

Hope you all had some fun.