Question

Parametric arc length?

I don't understand the part where the integral bounds switch to arctan. Why is this necessary?

The red marks mean "1" since the bounds are `0\let\le1` (not `2\pi`)

Answer 1

see explanation (thanks to Steve M.)

Explanation:

`\int_0^\color(blue)(1)\sqrt(1+\color(red)(t^2))\color(olive)(dt)`
`\int_0^\arctan(\color(blue)(1))\sqrt(1+\color(red)(\tan^2\theta))\color(olive)(\sec^2\thetad\theta)`