$$\text{Arc length}= \int_{t_1}^{t_2} \left|R'(t)\right|\,\mathrm dt$$
This looks similar to the formula for the magnitude of displacement, as the integral gives the area under the velocity-time graph.
But arc length, as I understand it, should be 'distance traveled', rather than the magnitude of the displacement vector. Because, for displacement, only the start and end points matter, while for distance traveled, the path taken also matters. Since arc length is the length measured along the curve, I feel it should be equal to distance traveled, rather than displacement.
What is that I'm missing here?