The 1, the Square, and the Root

You'll learnto rebuild the arc length formula from one triangle, to recognise which part is missing from a malformed integrand, to bound a length below by the chord, and to tell distance travelled 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 square root over the whole thing. Each omission produces a well-formed integral that evaluates to something plausible, so none of them announces itself. Derive the formula from a triangle once and all three stay put — because each one is a side of that triangle rather than a piece of notation to remember.

8 STEPS · 6 QUICK CHECKS · BC ONLY · REBUILD IT FROM THE TRIANGLE · v1

ds = √(dx² + dy²) ⇒ L = ∫ √(1 + (f′)²) dx ⇒ and the 1 was (dx/dx)²

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 horizontal leg is dx, its vertical leg is dy, and its hypotenuse is the piece of curve.

The question

How long is that hypotenuse, and what happens when you add up all of them across the interval?

Watch for

Every part of the formula comes from a side of this triangle. A formula recalled as a shape loses parts; one rebuilt from the triangle does not.

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