Step by step
The question
Question: Water flows into a tank at r(t) = t^3 + 1 litres per minute.
How much flows in from t = 0 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 0 to 2.
amount = ∫ from 0 to 2 of (t3 + 1) 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 t⁴ over 4 + t.
= (t44 + t) from 0 to 2
Put in 2, then take away the same with 0.
= (244 + 2) − (044 + 0)
That is 6 minus 0: 6.
= 6 − 0 = 6
So 6 litres flow in.
amount = 6 litres
Careful!
r is the rate, not the amount.
The amount is the integral of the rate.
Careful: not r(2) − r(0) ✗
2 Check
Let's check with a calculator's integral: about 6.
Right!
calculator: ∫ from 0 to 2 of r(t) dt ≈ 6
Let's look back.
First: the net change.
Last: check.
Answer
Careful: not r(2) − r(0) ✗