How do you find the displacement of a particle over the interval [a,b]?

How do you find the displacement of a particle over the interval [a,b]?

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

Answer: (B)

Displacement is the net change in position, which comes from velocity being the rate of change of position. Since v(t) = ds/dt, the change in position from time a to b is ∫_a^b v(t) dt, and by the Fundamental Theorem of Calculus this equals s(b) − s(a). So integrating velocity over the interval gives the displacement. The other options don’t match: integrating position over time wouldn’t measure a change in position; integrating acceleration would give velocity; integrating the absolute value of velocity would give total distance traveled, not displacement.

Displacement is the net change in position, which comes from velocity being the rate of change of position. Since v(t) = ds/dt, the change in position from time a to b is ∫_a^b v(t) dt, and by the Fundamental Theorem of Calculus this equals s(b) − s(a). So integrating velocity over the interval gives the displacement. The other options don’t match: integrating position over time wouldn’t measure a change in position; integrating acceleration would give velocity; integrating the absolute value of velocity would give total distance traveled, not displacement.