Accumulated Change in Context

Objective: Explain net change and initial amount, apply the stated method, and check its conditions.

If Q′ is an integrable net rate, Q(b)=Q(a)+∫ₐᵇ Q′(t)dt. The integral gives a change, not automatically the final amount. Include an initial value and preserve rate-times-time units when interpreting accumulated flow. An accumulated change and a final amount are different quantities.

Objective: Explain displacement and distance, apply the stated method, and check its conditions.

Integrating signed velocity gives displacement, while integrating its absolute value gives distance traveled. They coincide on intervals where velocity is nonnegative relative to the selected direction. If velocity changes sign, split or use absolute value to calculate total distance. Net displacement can be zero while total distance is positive.