LEARN · EXPLAIN · REVISE

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

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

TOPIC 01

Integration, area and school extensions

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

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

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

01 / Transfer#Your turn

t=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=\pi\int_0^t[y(x)]^2dx

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$V=\pi[x^3/3]_0^{14}=2744\pi/3$
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.

Teacher preparation and assessment

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?

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