Step by step
The question
Question: Water flows into a tank at r(t) = 3t^2 + 2 litres per minute.
How much flows in from t = 1 to t = 3?
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 3.
amount = ∫ from 1 to 3 of (3t2 + 2) 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 t³ + 2t.
= (t3 + 2t) from 1 to 3
Put in 3, then take away the same with 1.
= (33 + 2(3)) − (13 + 2(1))
That is 33 minus 3: 30.
= 33 − 3 = 30
So 30 litres flow in.
amount = 30 litres
Careful!
r is the rate, not the amount.
The amount is the integral of the rate.
Careful: not r(3) − r(1) ✗
2 Check
Let's check with a calculator's integral: about 30.
Right!
calculator: ∫ from 1 to 3 of r(t) dt ≈ 30
Let's look back.
First: the net change.
Last: check.
Answer
Careful: not r(3) − r(1) ✗