Concept preview

What you will be able to do

1. 01Formulate the "final amount = initial amount + accumulated change" relationship $F(b)=F(a)+\int_a^b F'(t)\,dt$ and use it to answer applied questions about tanks, costs, and populations.
2. 02Interpret a definite integral of a rate as the NET CHANGE of the accumulated quantity over an interval, and attach the correct units to that change.
3. 03Set up and evaluate accumulation problems from a rate given as a formula, a table, or a graph, without forgetting the initial value.

Suppose a quantity $F(t)$ changes over time and you know its rate of change $F'(t)$ — the derivative. The Fundamental Theorem of Calculus says that adding up the rate over an interval recovers the total change in the quantity: $\int_a^b F'(t)\,dt = F(b)-F(a)$. Read the right-hand side out loud: it is the ending value minus the starting value, which is precisely the net change in $F$ from $t=a$ to $t=b$. So the definite integral of a rate is a CHANGE, a difference, not the amount itself.