I ran into a rather abstruse question in today's first mechanics recitation. The question gave the one dimensional position of a particle with respect to time as $x = 10 - 4t + 2t^3$ It then asked for the distance traveled by the particle between t = 0 and t = 2. The suggested answer, (from the prof in charge of TAs), was to plot the trajectory of the particle, thereby demonstrating the distance and displacement were different. Here's the plot: The idea is that you can see that the particle travelled form 10 to 8 and then back to 18, so the total distance is more than the displacement from 10 to 18 i.e. 8. The question came up as to how to do this to get the exact answer. Here goes What we want to do is add up all the small, (read infinitesimal), distances travelled by the particle between time 0 and time 2. The phrases, 'adding up', and infinitesimal provide the tip off that we'll do an integral, so: Getting to the Integral the ...