A particle moves along a line with velocity meters per second for seconds
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
A particle moves along a line with velocity v(t)=4t3 meters per second for 0≤t≤2 seconds. Displacement over [0,2] is given by ∫024t3dt.
(a) Partition the interval [0,2] into N equal subintervals and choose the right endpoint of each subinterval as the sample time. Write the width Δt and the i-th right endpoint ti in terms of N and i.
(b) Derive an explicit limit expression (involving N→∞ and a single summation) whose value equals the displacement.
(c) Evaluate the limit from part (b) to obtain the exact displacement of the particle.
(d) Interpret your answer in the context of the particle’s motion, including appropriate units.