How do you find the total distance a particle travels on a given interval on a graph and net distance?

Answer 1

look up the arc length. The particle must move along a curve. At each small interval, the curve can be considered a straight line, with "arc length"

ds^2 = dx^2 + dy^2

Thus, adding up all the tiny hypotenuses gives.

s = ∫ds
= ∫√(dx^2+dy^2)
= ∫√(1+(y')^2) dx