The Net Change Theorem

Calculus 1 · Axiom Academy

Integrate a rate of change over an interval and you get the total change in the quantity — the bridge from derivatives back to integrals. 1. Integrate a Rate, Get the Net Change Suppose a quantity F changes over time, and you know its rate F'(x) at every instant. Add up that rate across [a,b] — that is exactly what the integral does — and you recover how much F changed in total : Watch a rate curve get swept left to right. The signed area it leaves behind is the accumulated change — and below the axis the rate is negative, so that area subtracts . The running total is the net change so far. 2. Net Change Is Signed: Displacement vs. Distance The most famous case: let v(t) be velocity. Then is the net change in position — the displacement . Crucially, it is signed : moving backward (negative velocity) eats into the total. Take v(t) = t^2 - 4 on [0,3] . It is negative until t=2 (the particle drifts left), then positive (it heads back right). Watch the particle move, and watch the two readouts diverge: The particle moves backward. The integral decreases — net change is going negative. The particle moves forward again, so climbs back up — but it does not fully recover. The signed net change in position: it ends at -3 . The particle finishes 3 units left of where it began. Total ground covered, ignoring direction: . Always at least the displacement.

This is the written version of the interactive lesson above. See the full Calculus 1 course.