Accumulated change | Year 12

01 / LEARN

Understand the idea.

Rate multiplied by time gives quantity

Integrating a rate gives the net change in its quantity over the chosen interval. To find a final amount, add that change to the amount at the start of the same interval. A production total asks only for newly produced items.

Use units to check the setup: items per hour integrated over hours gives items; litres per minute integrated over minutes gives litres. The integration bounds must match the period and the time unit used by the rate formula.

Q (b) = Q (a) + \int_{a}^{b} Q^{'} (t) d t

Net rate includes every contribution

When material both enters and leaves, form the net rate as inflow minus outflow. Its integral gives the change in contents. Total incoming material is a different quantity and requires integrating the inflow rate alone.

A rate model applies within its stated operating conditions. Check that the predicted amount remains physically possible. If a formula would empty a tank or exceed its capacity, its later values need a changed model rather than automatic extrapolation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

A workshop produces at $P^{'} (t) = 100 + 20 t$ items/hour for 0 to 4 hours. How many items are produced?

Worked solution

1
Set up new production over the stated time.
$Δ P = \int_{0}^{4} (100 + 20 t) d t$
The rate is measured in items per hour and the four-hour bounds give the number of new items.
2
Integrate the rate.
$Δ P = {[100 t + 10 t^{2}]}_{0}^{4}$
Constant production contributes a linear term; the increasing-rate term contributes a quadratic term.
3
Evaluate and interpret.
$Δ P = 400 + 160 = 560 items$
The question asks for production during the interval, so no starting inventory is added.

Answer

560 items

EXAMPLE 02

Worked example

A tank starts with 50 L. Inflow is $8 + t$ L/min and outflow is 3 L/min for $0 \leq t \leq 4$ . Find its final volume.

Worked solution

1
Form the net flow.
$V^{'} (t) = (8 + t) - 3 = 5 + t L/min$
Outflow decreases the contents and must be subtracted from inflow.
2
Accumulate the change over four minutes.
$Δ V = \int_{0}^{4} (5 + t) d t = {[5 t + \frac{t^{2}}{2}]}_{0}^{4}$
The integral of net flow gives the change in volume, not the final contents.
3
Evaluate the change.
$Δ V = 20 + 8 = 28 L$
The positive net rate produces a positive volume increase.
4
Add the initial contents.
$V (4) = 50 + 28 = 78 L$
The initial 50 litres were already present before the accumulated change.

Answer

78 L

03 / YOUR TURN

Try it yourself.

Check your understanding

A quantity starts at 12 and changes at $2 t$ units/min for 0 to 3 min. What is its final value?

Show answer & reasoning

$21$

The accumulated change is $\int_{0}^{3} 2 t d t = {[t^{2}]}_{0}^{3} = 9$ . Adding the starting amount gives $12 + 9 = 21$ units.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

A tank starts with 20 L and fills at $4 + 2 t$ L/min for 0 to 3 min. Find the final volume.
3 marks
A machine uses power $P (t) = 3 t^{2}$ kW for 0 to 2 hours. Find the energy used.
3 marks
A stock starts at 100 units and changes at $6 - 2 t$ units/day for 0 to 4 days. Find its net change and final size.
3 marks
An inflow is 10 L/min and an outflow is $t$ L/min for 0 to 5 min. Find the total inflow and net change.
3 marks
A population model has $N^{'} (t) = 2 e^{t}$ and $N (0) = 30$ . Find $N (ln 3)$ .
3 marks
A rate model predicts that a 200 L tank contains 230 L after 10 minutes. What needs checking before accepting that result?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A tank starts with 20 L and fills at $4 + 2 t$ L/min for 0 to 3 min. Find the final volume.
3 marks
Change $= {[4 t + t^{2}]}_{0}^{3} = 21$ L.
Final volume $20 + 21 = 41$ L.
41 L
A machine uses power $P (t) = 3 t^{2}$ kW for 0 to 2 hours. Find the energy used.
3 marks
Energy $= \int_{0}^{2} 3 t^{2} d t = 8$ .
8 kWh
A stock starts at 100 units and changes at $6 - 2 t$ units/day for 0 to 4 days. Find its net change and final size.
3 marks
Change $= {[6 t - t^{2}]}_{0}^{4} = 24 - 16 = 8$ .
Add 100.
Net change 8 units; final 108 units.
An inflow is 10 L/min and an outflow is $t$ L/min for 0 to 5 min. Find the total inflow and net change.
3 marks
Inflow $= 10 (5) = 50$ .
Net change $= \int_{0}^{5} (10 - t) d t = 50 - 12.5 = 37.5$ .
50 L inflow;37.5 L net gain.
A population model has $N^{'} (t) = 2 e^{t}$ and $N (0) = 30$ . Find $N (ln 3)$ .
3 marks
Change $= {[2 e^{t}]}_{0}^{ln 3} = 6 - 2 = 4$ .
Add 30.
$34$
A rate model predicts that a 200 L tank contains 230 L after 10 minutes. What needs checking before accepting that result?
3 marks
The capacity is exceeded; overflow or a changed flow rule must be modelled.
Whether the model remains valid after the tank fills.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods integration and logarithmic antiderivatives; original learning and assessment material, with by-hand methods and explicitly described numerical models.

Official curriculum source ↗

In this lesson

Understand the idea.

Rate multiplied by time gives quantity

Net rate includes every contribution

See the method in action.

Worked example

Set up new production over the stated time.

Integrate the rate.

Evaluate and interpret.

Worked example

Form the net flow.

Accumulate the change over four minutes.

Evaluate the change.

Add the initial contents.

Try it yourself.

Check your understanding

Keep practising