Accumulated change | Year 12

01 / LEARN

The idea, step by step

Rate multiplied by time gives quantity

A rate says how quickly something changes. Integrating the rate over an interval gives the net change during that interval. If the question asks for the final amount, add the initial amount; an integral by itself does not include what was there at the start.

The independent variable determines the units. Integrating dollars per item with respect to items gives dollars; integrating litres per hour with respect to hours gives litres. Read the rate definition before choosing the limits.

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

Net rate includes every contribution

When a tank both fills and drains, the net rate is inflow minus outflow. It may be negative even though both individual flow rates are positive. Integrate the net rate for the change in contents; integrate the inflow alone for total volume supplied.

Check physical limits and the stated model interval. A formula predicting negative stock or a volume above the tank capacity cannot be used unchanged beyond that event. Explain the limitation rather than accepting every algebraic output.

A common trap

Giving the accumulated change when the question asks for the total amount.

Instead: Write a starting-amount-plus-change equation, then evaluate.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

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

Production $= \int_{0}^{4} (100 + 20 t) d t$ .
${[100 t + 10 t^{2}]}_{0}^{4} = 400 + 160 = 560$ .
No starting stock is needed because the question asks only for new production.

560 items

Example 2

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.

Net rate $= (8 + t) - 3 = 5 + t$ .
Change $= \int_{0}^{4} (5 + t) d t = 20 + 8 = 28$ L.
Add initial 50 L to obtain 78 L.

78 L

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

$21$

Change $= \int_{0}^{3} 2 t d t = 9$ ; add 12.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF 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.

Checking saved progress…

Continue learning

Displacement and total distance

Next lesson

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 ↗

Back to Integration