Which integral gives the total distance traveled by a particle over the interval [a,b]?

Which integral gives the total distance traveled by a particle over the interval [a,b]?

• A. ∫_a^b a(t) dt
• B. ∫_a^b s(t) dt
• C. ∫_a^b v(t) dt
• D. ∫_a^b |v(t)| dt

Answer: (D)

The distance a particle travels depends on how much ground it covers, ignoring direction. That means you use the speed, which is the magnitude of velocity, and accumulate it over time. Mathematically, the total distance from time a to b is the integral of the speed: ∫ from a to b of |v(t)| dt. If you simply integrate velocity, you get displacement, which can be smaller than the distance traveled when the particle reverses direction, because negative velocity portions subtract. For example, with v(t) = sin t on [0, 2π], the net displacement is zero, but the distance traveled is 4. Other integrals like those of acceleration or position over time don’t measure distance traveled. Hence the integral of the absolute value of velocity gives the total distance.

The distance a particle travels depends on how much ground it covers, ignoring direction. That means you use the speed, which is the magnitude of velocity, and accumulate it over time. Mathematically, the total distance from time a to b is the integral of the speed: ∫ from a to b of |v(t)| dt. If you simply integrate velocity, you get displacement, which can be smaller than the distance traveled when the particle reverses direction, because negative velocity portions subtract. For example, with v(t) = sin t on [0, 2π], the net displacement is zero, but the distance traveled is 4. Other integrals like those of acceleration or position over time don’t measure distance traveled. Hence the integral of the absolute value of velocity gives the total distance.