Step by step
The question
Question: Sand flows into a pile at r(t) = 16 - t² cubic metres 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 rule: the amount is the integral of the rate.
amount = ∫ from a to b of r(t) dt
Here it goes from 0 to 3.
= ∫ from 0 to 3 of (16 − t2) dt
The antiderivative: add 1 to each power, divide by the new power.
Now we have 16t − t³ over 3.
= (16t − t33) from 0 to 3
Put in 3, then take away the same with 0.
= (16(3) − 333) − (16(0) − 033)
That is 39 minus 0: 39.
= 39 − 0 = 39
So 39 cubic metres flow in.
amount = 39 cubic metres
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 39.
Right!
calculator: ∫ from 0 to 3 of r(t) dt ≈ 39
Let's look back.
First: the net change.
Last: check.
Answer
amount = 39 cubic metres