Step by step
The question
Question: Air flows into a balloon at r(t) = 2t + 10 litres per second.
How much flows in from t = 2 to t = 6?
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 2 to 6.
amount = ∫ from 2 to 6 of (2t + 10) 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 t² + 10t.
= (t2 + 10t) from 2 to 6
Put in 6, then take away the same with 2.
= (62 + 10(6)) − (22 + 10(2))
That is 96 minus 24: 72.
= 96 − 24 = 72
So 72 litres flow in.
amount = 72 litres
Careful!
r is the rate, not the amount.
The amount is the integral of the rate.
Careful: not r(6) − r(2) ✗
2 Check
Let's check with a calculator's integral: about 72.
Right!
calculator: ∫ from 2 to 6 of r(t) dt ≈ 72
Let's look back.
First: the net change.
Last: check.
Answer
Careful: not r(6) − r(2) ✗