Kenneth ChengMathematical modeling
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Integration, area and school extensions

Read the idea, work independently, then explain what changed.

2027 JM02 考試大綱 · 6. Basic calculus: integrals and area · PDF 2 / printed page 2

Revisit first: Derivative operations

TOPIC 01

Integration, area and school extensions

Find antiderivatives, evaluate definite integrals and distinguish signed accumulation from geometric area.

What you will be able to explain

  • Find antiderivatives, evaluate definite integrals and distinguish signed accumulation from geometric area.
  • Justify the method and check the conditions in a new situation.

Model or definition

Find antiderivatives, evaluate definite integrals and distinguish signed accumulation from geometric area.

∫abf(x)dx=F(b)−F(a)\int_a^bf(x)dx=F(b)-F(a)

Conditions

An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Can the signed integral be zero while the total area is positive?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

y=ax²; tangent slope at x=t is 2; signed integral from 0 to t is 0.3333. Signed integral can be negative.

y=ax²; tangent slope at x=t is 2; signed integral from 0 to t is 0.3333. Signed integral can be negative.

Explain: Compare two admissible cases and explain their different results using the stated model.

Transfer: Compare x on a symmetric interval with |x| on that interval.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Find all antiderivatives of tx².

t=2t=2
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Raise the power and divide by the new exponent.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    d(2x3/3+C)/dx=2x2d(2x^3/3+C)/dx=2x^2

  3. The derivative check confirms the power and constant.

The requested relation or conclusion is shown below.

2x3/3+C2x^3/3+C

Checks and common pitfalls: The derivative check confirms the power and constant.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Find the area between y=t and y=x for 0≤x≤t.

t=3t=3
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
A=∫0t(t−x)dxA=\int_0^t(t-x)dx
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    A=[3x−x2/2]03=9/2A=[3x-x^2/2]_0^{3}=9/2

  3. Top minus bottom is nonnegative throughout this interval.

The requested value is 4.5.

Checks and common pitfalls: Top minus bottom is nonnegative throughout this interval.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

School extension: revolve y=x, 0≤x≤t, about the x-axis. Use the disk model to find volume.

t=4t=4
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
V=π∫0t[y(x)]2dxV=\pi\int_0^t[y(x)]^2dx
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    V=π[x3/3]04=64π/3V=\pi[x^3/3]_0^{4}=64\pi/3

  3. The squared radius enters disk area; integrating y alone would give area rather than volume.

The requested value is 67.0206432766.

Checks and common pitfalls: The squared radius enters disk area; integrating y alone would give area rather than volume.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Find all antiderivatives of tx².

t=5t=5
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Raise the power and divide by the new exponent.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    d(5x3/3+C)/dx=5x2d(5x^3/3+C)/dx=5x^2

  3. The derivative check confirms the power and constant.

The requested relation or conclusion is shown below.

5x3/3+C5x^3/3+C

Checks and common pitfalls: The derivative check confirms the power and constant.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Evaluate the definite integral.

∫062x dx\int_0^{6}2x\,dx
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
F(x)=x2F(x)=x^2
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    F(6)−F(0)=36F(6)-F(0)=36

  3. A definite integral has a value, not an arbitrary integration constant.

The requested value is 36.

Checks and common pitfalls: A definite integral has a value, not an arbitrary integration constant.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Evaluate the signed integral.

∫−77x dx\int_{-7}^7x\,dx
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use odd symmetry or the antiderivative.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    [x2/2]−77=0[x^2/2]_{-7}^7=0

  3. The two signed contributions cancel.

The requested value is 0.

Checks and common pitfalls: The two signed contributions cancel.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Find the geometric area between y=x and the x-axis on [−t,t].

t=8t=8
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
Area=∫−t0(−x)dx+∫0txdxArea=\int_{-t}^0(-x)dx+\int_0^txdx
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    A=64/2+64/2=64A=64/2+64/2=64

  3. Use the absolute value of the integrand for area.

The requested value is 64.

Checks and common pitfalls: Use the absolute value of the integrand for area.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Find the geometric area between y=x and the x-axis on [−t,t].

t=9t=9
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
Area=∫−t0(−x)dx+∫0txdxArea=\int_{-t}^0(-x)dx+\int_0^txdx
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    A=81/2+81/2=81A=81/2+81/2=81

  3. Use the absolute value of the integrand for area.

The requested value is 81.

Checks and common pitfalls: Use the absolute value of the integrand for area.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Find the area between y=t and y=x for 0≤x≤t.

t=10t=10
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
A=∫0t(t−x)dxA=\int_0^t(t-x)dx
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    A=[10x−x2/2]010=100/2A=[10x-x^2/2]_0^{10}=100/2

  3. Top minus bottom is nonnegative throughout this interval.

The requested value is 50.

Checks and common pitfalls: Top minus bottom is nonnegative throughout this interval.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

F′(x)=2x and F(0)=t. Find F(2).

t=11t=11
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
F(x)=x2+CF(x)=x²+C
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    C=11C=11

  3. Apply the stated relation and retain its conditions.

    F(2)=4+11=15F(2)=4+11=15

  4. An initial condition fixes the integration constant.

The requested value is 15.

Checks and common pitfalls: An initial condition fixes the integration constant.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

School extension: revolve y=x, 0≤x≤t, about the x-axis. Use the disk model to find volume.

t=12t=12
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
V=π∫0t[y(x)]2dxV=\pi\int_0^t[y(x)]^2dx
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    V=π[x3/3]012=1728π/3V=\pi[x^3/3]_0^{12}=1728\pi/3

  3. The squared radius enters disk area; integrating y alone would give area rather than volume.

The requested value is 1809.55736847.

Checks and common pitfalls: The squared radius enters disk area; integrating y alone would give area rather than volume.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

F′(x)=2x and F(0)=t. Find F(2).

t=13t=13
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
F(x)=x2+CF(x)=x²+C
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    C=13C=13

  3. Apply the stated relation and retain its conditions.

    F(2)=4+13=17F(2)=4+13=17

  4. An initial condition fixes the integration constant.

The requested value is 17.

Checks and common pitfalls: An initial condition fixes the integration constant.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

School extension: revolve y=x, 0≤x≤t, about the x-axis. Use the disk model to find volume.

t=14t=14
  • An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
V=π∫0t[y(x)]2dxV=\pi\int_0^t[y(x)]^2dx
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    V=π[x3/3]014=2744π/3V=\pi[x^3/3]_0^{14}=2744\pi/3

  3. The squared radius enters disk area; integrating y alone would give area rather than volume.

The requested value is 2873.51008048.

Checks and common pitfalls: The squared radius enters disk area; integrating y alone would give area rather than volume.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • Find antiderivatives, evaluate definite integrals and distinguish signed accumulation from geometric area.
    • Which condition is essential in integration, area and school extensions?
    • Can the signed integral be zero while the total area is positive?

    Board plan

    • Model or definition: Find antiderivatives, evaluate definite integrals and distinguish signed accumulation from geometric area.
      ∫abf(x)dx=F(b)−F(a)\int_a^bf(x)dx=F(b)-F(a)
    • Conditions: An indefinite integral includes an arbitrary constant; split at sign changes when calculating area. Revolution volumes are a school extension (T06 PDF6–10).

    Anticipated thinking

    • A negative definite integral is not a negative geometric area.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

    No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

    Curriculum and source notes ↗