How many litres of water are pumped into the tank from t = 0 to t = 3?

Step by step, with the reason for every step · AP Calculus AB Practice Exam 1 · AP Calculus AB

Step by step

Question: Water flows into a tank at r(t) = 3t² + 2t litres per minute.

How much flows in from t = 0 to t = 3?

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 (3t² + 2t) dt

The antiderivative: add 1 to each power, divide by the new power.

Now we have t³ + t².

= (t³ + t²) from 0 to 3

Put in 3, then take away the same with 0.

= (3³ + 3²) − (0³ + 0²)

That is 36 minus 0: 36.

= 36 − 0 = 36

So 36 litres flow in.

amount = 36 litres

Careful!

r is the rate, not the amount.

The amount is the integral of the rate.

Careful: not r(3) − r(0) ✗

Let's check with a calculator's integral: about 36.

Right!

calculator: ∫ from 0 to 3 of r(t) dt ≈ 36

Let's look back.

First: the net change.

Last: check.

Answer

amount = 36 litres

Try this one yourself, then check your answer:

Water flows into a tank at r(t) = 5t^2 + 6t litres per minute. How much flows in from t = 0 to t = 3?

Show the answer

72 litres