The 1, the Square, and the Root

You'll learnto rebuild the arc length formula from one triangle, to recognize which part is missing from a malformed integrand, to bound a length below by the chord, and to tell distance traveled from displacement on a parametric path.

Arc length has one formula and three parts that go missing separately: the added 1, the square on the derivative, and the root over the whole thing. Each omission still gives a well-formed integral with a plausible answer, so none of them announces itself. Build the formula from one triangle and all three stay put, because each is a side of that triangle rather than a piece of notation. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · BC ONLY · REBUILD IT FROM THE TRIANGLE · v2

Every part of the formula is a side of one small triangle

Before you start

What you're looking at

A curve with a small right triangle sitting on a short piece of it, and the same triangle magnified beside it. Its legs are a step across and a step up; its hypotenuse is the piece of curve.

The question

How long is that hypotenuse, and what do you get by adding up all of them across the interval?

Step 1 / 12