The Net Change Theorem

Definition

The Net Change Theorem is a physical application of the Fundamental Theorem of calculus. It states that the definite integral of a rate of change is the total net change in the quantity over that time interval.

Why It Matters

The integral of a rate of change is the net change. In economics, this links marginal cost to total cost; in physics, it links velocity to displacement. If you ignore this theorem, you lose the ability to reconstruct the state of a system from its growth patterns.

Core Concepts

General Form: $\int_a^b Q'(t) dt = Q(b) - Q(a)$ .
- How to read: “The integral from a to b of the derivative of Q with respect to t is equal to Q evaluated at b minus Q evaluated at a.”
- Meaning / when to use: Integrating a rate of change over an interval gives the net change in the quantity. This is FTC Part 2 applied to real-world accumulation problems.
Displacement vs. Distance:
- Displacement: $\int_{t_1}^{t_2} v(t) dt$ $\int_{t_{1}}^{t_{2}} v (t) d t$ (net change in position).
  - How to read: “The integral from time t one to time t two of the velocity with respect to time.”
  - Meaning: Signed net movement — forward minus backward. Can be zero if you return to start.
- Total Distance: $\int_{t_1}^{t_2} |v(t)| dt$ $\int_{t_{1}}^{t_{2}} ∣ v (t) ∣ d t$ (sum of all movement, regardless of direction).
  - How to read: “The integral from time t one to time t two of the absolute value of the velocity with respect to time.”
  - Meaning: Always non-negative; counts every meter traveled, ignoring direction reversals.
Broad Applicability: Works for any rate—marginal cost to total cost, flow rate to total volume, reaction rate to total substance concentration.

The Net Change Theorem

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes