Arc length when the curve intersect itself

Question

I came a cross a theorem saying that the arc length of a smooth non self-intersecting parametric curve is given by

$ L= \int_{0}^{2\pi} \sqrt(y’^2+x’^2)dx$

Why we specify that the curve should be smooth and not intersecting itself is that means we should split the interval and integrate each loop separately

Also if a particle travelled arround a circle 3 times from t=0 to $t=3\pi$ when we find the arc length is it giving us the whole distance that the particle travelled or it is only the length of the circle once?

Answer 1

If it is not smooth, the derivatives are undefined.

If it is self-intersecting, you can run around the loop forever.