Step by step
The question
Question: Water flows into a pool at r(t) = 12t^2 + 4t litres per minute.
How much flows in from t = 1 to t = 2?
1 The net change
r(t) is a rate: how fast it flows in.
The total that flows in is the integral of the rate, from 1 to 2.
amount = ∫ from 1 to 2 of (12t2 + 4t) dt
amount = ∫ from a to b of r(t) dt
The antiderivative: add 1 to each power, divide by the new power.
Now we have 4 t³ + 2 t².
= (4t3 + 2t2) from 1 to 2
Put in 2, then take away the same with 1.
= (4(2)3 + 2(2)2) − (4(1)3 + 2(1)2)
That is 40 minus 6: 34.
= 40 − 6 = 34
So 34 litres flow in.
amount = 34 litres
Careful!
r is the rate, not the amount.
The amount is the integral of the rate.
Careful: not r(2) − r(1) ✗
2 Check
Let's check with a calculator's integral: about 34.
Right!
calculator: ∫ from 1 to 2 of r(t) dt ≈ 34
Let's look back.
First: the net change.
Last: check.
Answer
Careful: not r(2) − r(1) ✗