Step by step
The question
Question: Water flows into a tank at r(t) = 4t + 5 litres per minute.
How much flows in from t = 0 to t = 4?
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 0 to 4.
amount = ∫ from 0 to 4 of (4t + 5) 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 2 t² + 5t.
= (2t2 + 5t) from 0 to 4
Put in 4, then take away the same with 0.
= (2(4)2 + 5(4)) − (2(0)2 + 5(0))
That is 52 minus 0: 52.
= 52 − 0 = 52
So 52 litres flow in.
amount = 52 litres
Careful!
r is the rate, not the amount.
The amount is the integral of the rate.
Careful: not r(4) − r(0) ✗
2 Check
Let's check with a calculator's integral: about 52.
Right!
calculator: ∫ from 0 to 4 of r(t) dt ≈ 52
Let's look back.
First: the net change.
Last: check.
Answer
Careful: not r(4) − r(0) ✗