Step by step
The question
Question: Oil flows into a tank at r(t) = 6t^2 gallons per hour.
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 6t2 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 t³.
= (2t3) from 0 to 2
Put in 2, then take away the same with 0.
= (2(2)3) − (2(0)3)
That is 16 minus 0: 16.
= 16 − 0 = 16
So 16 gallons flow in.
amount = 16 gallons
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 16.
Right!
calculator: ∫ from 0 to 2 of r(t) dt ≈ 16
Let's look back.
First: the net change.
Last: check.
Answer
Careful: not r(2) − r(0) ✗