A particle moves in the plane from time to with position vector
MATHEMATICS · Single correct · AP · Calculus BC · Applications of Integration · Applications of Integration
Problem
A particle moves in the plane from time t=a to t=b with position vector r(t). Which of the following statements best explains why the value of ∫ab∣v(t)∣dt is always greater than or equal to ∣r(b)−r(a)∣?
(a) Because moving the absolute value outside the integral would give ∫abv(t)dt, and keeping it inside must therefore produce a larger value.
(b) Because the integrand ∣v(t)∣ is always positive, whereas ∣r(b)−r(a)∣ can be negative if the particle reverses direction.
(c) Because the integral accumulates the entire length of the path traveled, while ∣r(b)−r(a)∣ measures only the straight-line distance between the endpoints.
(d) Because speed is the derivative of distance and integrating a derivative necessarily produces a value larger than the net change in position.