AP Calculus AB and BC glossary

Net change theorem

The net change theorem says that integrating a rate of change over an interval gives the net change in the underlying quantity. It is the interpretation of the Fundamental Theorem of Calculus that applied problems actually use.

∫abF′(t) dt=F(b)−F(a)\int_a^b F'(t)\,dt = F(b) - F(a)

This is what turns a rate into a total. If water flows into a tank at r(t)r(t) litres per minute, then ∫010r(t) dt\int_0^{10} r(t)\,dt is the number of litres added over those ten minutes, not the flow rate at any instant.

To get the final amount rather than the change, add the starting amount: F(b)=F(a)+∫abF′(t) dtF(b) = F(a) + \int_a^b F'(t)\,dt. That form answers most tank and population free-response questions directly.

Units

The integral of a rate has units of rate times the variable of integration, so litres per minute integrated over minutes gives litres. Checking units catches setup errors quickly.

Appears in: Unit 6: Integration and Accumulation, Unit 8: Applications of Integration