Rates and integration

Finding a quantity from its rate of change by integration, total change as the area under a rate graph, and rates involving exponential functions such as a flow that dies away.

Total change

Example 1: filling a tank●●

Water flows into a tank at $\dfrac{dV}{dt} = 12 - 3t$ L/min until the flow stops. The tank held $100$ L at the start. How much does it hold when the flow stops?

The flow stops when $12 - 3t = 0$, at $t = 4$.

$V = 12t - \tfrac32 t^2 + C$, and $V(0) = 100$ gives $C = 100$.

$V(4) = 48 - 24 + 100 = 124$ L.

Example 2: a leak●●

A tank holding $500$ L leaks at $20 - 2t$ L/min for $0 \le t \le 10$. How much is left at $t = 10$?

Water lost $= \displaystyle\int_0^{10} (20 - 2t)\,dt = 200 - 100 = 100$ L.

$500 - 100 = 400$ L remain.

A rate that dies away

Flow rate R = 50e^(−0.1t)

The shaded area is the total amount that has flowed. It approaches 500 m³ but never reaches it.

Example 3●●●

After a storm, water flows out of a dam at $R = 50e^{-0.1t}$ m³/h. How much flows out in the first $10$ hours?

$\displaystyle\int_0^{10} 50e^{-0.1t}\,dt = \Big[-500e^{-0.1t}\Big]_0^{10} = 500\left(1 - e^{-1}\right)$.

That is about $316$ m³. Over a very long time the total approaches $500$ m³.

Flashcards

Total change from a rate

$\displaystyle\int_a^b R\,dt$

Geometric meaning

The signed area under the rate graph.

A negative rate

The quantity is decreasing.

Practice

In Example 1, how much water is in the tank when the flow stops?
In Example 2, how much water is left at $t = 10$?
In Example 3, how much water flows out in the first $10$ hours, to the nearest m³?
In Example 3, the total amount that flows out over a very long time approaches:
$\dfrac{dV}{dt} = 6t$ and $V(0) = 10$. Find $V(2)$.
$\dfrac{dV}{dt} = 3t^2 - 12$ and $V(0) = 20$. Find $V(3)$.
A population grows at $\dfrac{dP}{dt} = 200e^{0.05t}$, with $P(0) = 4000$. Find $P(10)$ to the nearest whole number.
The total change in a quantity over an interval equals:
On an interval where the rate $\dfrac{dQ}{dt}$ is negative: