Step by step
The question
Question: Oil flows into a tank at r(t) = 8 - 2t gallons per hour.
How much flows in from t = 0 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 0 to 3.
amount = ∫ from 0 to 3 of (8 − 2t) 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 8t − t².
= (8t − t2) from 0 to 3
Put in 3, then take away the same with 0.
= (8(3) − 32) − (8(0) − 02)
That is 15 minus 0: 15.
= 15 − 0 = 15
So 15 gallons flow in.
amount = 15 gallons
Careful!
r is the rate, not the amount.
The amount is the integral of the rate.
Careful: not r(3) − r(0) ✗
2 Check
Let's check with a calculator's integral: about 15.
Right!
calculator: ∫ from 0 to 3 of r(t) dt ≈ 15
Let's look back.
First: the net change.
Last: check.
Answer
Careful: not r(3) − r(0) ✗