Interpreting a result in context | Year 12

01 / LEARN

The idea, step by step

Interpret the integrand and the differential

An integral combines a rate with a change in its input variable. A flow in litres per minute integrated over minutes gives litres. A cross-sectional area in square metres integrated along metres gives cubic metres. Units help check whether the chosen model matches the question.

An accumulated change is not automatically a final amount. Add the initial amount when the question asks for the final quantity. State the interval and the model assumptions; predictions outside the model's sensible range need caution.

Check the result in context

The average value of f over [a,b] is $\frac{1}{b - a} \int_{a}^{b} f (x) d x$ . It has the same units as f, unlike the raw integral. Average speed and average signed velocity are different when motion reverses.

Use the graph, a rough bound and an independent derivative check to judge plausibility. Report whether an exact or rounded result is useful, including units and the meaning of the answer rather than only a number.

A common trap

Integrating a rate always gives the final amount.

Instead: It gives the change over the interval. Include the starting amount when a final amount is requested.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

A pipe has cross-sectional area A(x)=4x+1 square metres for 0<=x<=3 metres. Find its volume.

Volume is the integral of A along the pipe.
${[2 x^{2} + x]}_{0}^{3} = 21$ .

21 m^{3}

Example 2

Worked example 2

A force F(x)=3x squared+2 N acts along a straight displacement from x=0 to 2 m. Find the work.

Work is $\int_{0}^{2} (3 x^{2} + 2) d x$ .
Evaluate ${[x^{3} + 2 x]}_{0}^{2} = 12$ .

12 J

Check your understanding

A rate in metres per second is integrated over seconds. What are the resulting units?

Show answer

Metres.

The time units cancel, leaving displacement units.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Water enters at 2t+3 litres/minute for 0<=t<=4. Initially there are 10 litres. Find the final amount.
2 marks
Find the average temperature 20+2t degrees C over 0<=t<=4 hours.
2 marks
Velocity is v=t-2 m/s for 0<=t<=4. Find displacement and total distance.
2 marks
A population increases at 100 exp(0.1t) people/year for 0<=t<=10. Find the increase.
2 marks
Marginal cost is 6q+10 dollars/unit. Find the cost increase when output rises from q=0 to 5.
2 marks
A solid has cross-section area pi(1+x) squared for 0<=x<=2 metres. Find its volume.
2 marks

Read the worked solutions

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

Water enters at 2t+3 litres/minute for 0<=t<=4. Initially there are 10 litres. Find the final amount.
2 marks
The added amount is ${[t^{2} + 3 t]}_{0}^{4} = 28$ .
Add the initial 10 litres.
$38$ litres
Find the average temperature 20+2t degrees C over 0<=t<=4 hours.
2 marks
Integrate to obtain 96 degree-hours.
Divide by the 4-hour interval.
$24^{\circ} C$ .
Velocity is v=t-2 m/s for 0<=t<=4. Find displacement and total distance.
2 marks
The signed integral cancels.
Split at t=2 and integrate speed on each side.
Displacement 0 m; distance 4 m.
A population increases at 100 exp(0.1t) people/year for 0<=t<=10. Find the increase.
2 marks
An antiderivative is 1000 exp(0.1t).
Subtract its t=0 value from its t=10 value.
$1000 (e - 1)$ people, approximately 1718.
Marginal cost is 6q+10 dollars/unit. Find the cost increase when output rises from q=0 to 5.
2 marks
Integrate marginal cost over output.
${[3 q^{2} + 10 q]}_{0}^{5} = 125$ .
$125$ .
A solid has cross-section area pi(1+x) squared for 0<=x<=2 metres. Find its volume.
2 marks
Integrate pi times $(1 + 2 x + x^{2})$ .
Evaluate to pi times $(2 + 4 + 8 / 3)$ .
$26 π / 3 m^{3}$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics integration techniques and applications, including inverse-trigonometric patterns, substitution, parts, simple partial fractions, areas and solids of revolution.

Official curriculum source ↗

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