The Net Change Theorem connects derivatives and integrals in one of the most useful ideas in calculus. If a derivative tells you how fast a quantity is changing, then the integral of that derivative tells you how much the quantity changed overall. This matters in physics, economics, biology, and engineering because many real problems give rates first and ask for totals later.
The theorem is a direct application of the Fundamental Theorem of Calculus.
Signed area is the key visual idea behind the theorem. Area above the x-axis adds positive change, while area below the x-axis adds negative change, so the integral gives net change rather than total amount traveled. For motion, integrating velocity gives displacement, but integrating speed gives total distance.
This difference explains why a car can travel many meters while ending up with a small or even zero displacement.
Key Facts
- Net Change Theorem: integral from a to b of F'(x) dx = F(b) - F(a).
- Plain meaning: integrating a rate of change gives the total net change in the original quantity.
- If v(t) is velocity, then integral from a to b of v(t) dt = s(b) - s(a), the displacement.
- If speed is |v(t)|, then total distance traveled = integral from a to b of |v(t)| dt.
- Signed area above the x-axis is positive, and signed area below the x-axis is negative.
- Units of an integral of a rate are rate units times input units, such as meters per second times seconds = meters.
Vocabulary
- Net Change Theorem
- A theorem stating that the integral of a rate of change over an interval equals the change in the original quantity over that interval.
- Derivative
- A derivative measures the instantaneous rate at which one quantity changes with respect to another.
- Definite Integral
- A definite integral gives the signed accumulation of a function over a specific interval.
- Displacement
- Displacement is the net change in position, including direction, from the starting point to the ending point.
- Total Distance
- Total distance is the full length of a path traveled, found by accumulating speed rather than signed velocity.
Common Mistakes to Avoid
- Treating net change as total distance is wrong because negative velocity or negative rate values subtract from the signed integral.
- Forgetting the constant of integration is wrong in an indefinite integral, but in a definite net change calculation the result F(b) - F(a) does not need an added constant.
- Ignoring units is wrong because the integral of a rate has accumulated units, such as liters per minute times minutes giving liters.
- Using F(b) - F(a) when the graph shows F(x) instead of F'(x) is wrong because the Net Change Theorem applies to the integral of the rate of change, not automatically to the original function graph.
Practice Questions
- 1 A tank is filled at a rate r(t) = 4t + 3 liters per minute for 0 <= t <= 5. How many liters of water are added during the 5 minutes?
- 2 A particle moves with velocity v(t) = t^2 - 4t meters per second for 0 <= t <= 5. Find its displacement over the interval.
- 3 A velocity graph is above the time axis from t = 0 to t = 3 and below the time axis from t = 3 to t = 6. Explain why the integral of velocity may be smaller than the total distance traveled.