I am currently studying about Parametric integration and very confuse about the arc length meaning.
In my understanding, the arc length is the total length that an object traveled along a curve in a given interval, in other words the length of the curve in that interval. Let say an object travels on a circle motion with the parametric equations.
$x=3\cos(3t)$. $y=\sin(3t)$
in the interval $[0,2\pi]$
using the formula for arc length which is $ L= \int_{0}^{2\pi} \sqrt(y’^2+x’^2)dx$ the answer is $18\pi$ which meets my understanding that arc length is the whole distance travelled so 3 times around the circle.
However if we given the parametric curve $x=\sin^2(t)$ $y=\cos^2(t)$ for the interval $[0,3\pi]$
although the object should be going 6 times back and fourth between 2 points however when using L formula with limits 0 and $3\pi$ the answer is different from when we integrate in the interval $[0,\pi/2]$ and then multiply by 6, so why in this case the arc length formula is not working if we integrate over the full interval $[0,3\pi]$ as I thought that it would give us the total distance without changing the limits.
Also is my understanding correct that the formula would give the total distance traveled and not the displacement
$$\int_0^{\pi/2}\sqrt{4\sin^2t\cos^2x+4\cos^2t\sin^2t}\,dt=\sqrt2\int_0^{\pi/2}|\sin 2t|\,dt.$$ and using periodicity and symmetry, $$\int_0^{3\pi}|\sin 2t|\,dx=3\int_0^{\pi}|\sin 2t|\,dx=6\int_0^{\pi/2}|\sin 2t|\,dt.$$