Question

A particle moves along the x-axis with velocity v(t) = t^2 − 1, with t measured in seconds and v(t) measured in feet per second. Find the total distance travelled by the particle from t = 0 to t = 2 seconds.

Answer 1

clearly that would be

∫[0,2] t^2-1 dt = 2/3

But that is the algebraic distance (final displacement), since it starts out with negative velocity. To get the actual distance traveled, you need to separate the two parts:

∫[0,1] -(t^2-1) dt + ∫[1,2] t^2-1 dt = 2/3 + 4/3 = 2

Answer 2

uh my answer is .667