Using Accumulation Functions and Definite Integrals in Applied Contexts

Start here, add the change, land there

• Many applied problems give a rate and a starting amount, and ask for a later total.
• The key relationship: final value = initial value + accumulated change.
• The accumulated change is the definite integral of the rate over the interval.
• This one idea powers tanks, populations, accounts, and more.

The accumulation formula

• If $R(t)$ is the rate and $Q$ the quantity, then:
• $$Q(b)=Q(a)+\int_a^b R(t)\,dt$$
• Start with $Q(a)$, add the net accumulated change $\displaystyle\int_a^b R(t)\,dt$, arrive at $Q(b)$.
• It's the FTC in applied clothing: the integral of a rate is the net change of its quantity.

The area under a rate graph is the total amount accumulated

Reading the context and units

• A definite integral 定积分 of a rate has units of (rate units)$\times$(time) = the quantity's units.
• $\int$ of litres/min over minutes gives litres; $\int$ of people/year over years gives people.
• The sign matters: a negative rate (draining, cooling) subtracts from the total.
• Always interpret the integral in the problem's own words and units.

Net vs. gross change

• The integral $\int_a^b R\,dt$ is the net change — inflow minus outflow.
• If a tank is filled and drained, the net integral accounts for both automatically.
• To find total added or total removed separately, integrate only the positive or negative parts.
• But "final = initial + net accumulated change" always uses the plain signed integral.

Write from a final amount

• A tank starts with 20 litres.
• The net rate adds 7 litres over the interval.