Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Subtract inner disk area from outer disk area before integrating.
- Justify the conclusion "Volume=π∫₀¹(x²−x⁴)dx=2π/15" using the stated assumptions.
Before you start
Definite integration and disk area.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Rotate the region between y=x and y=x² for 0≤x≤1 about the x-axis. Find its volume.
Why this math matters
Subtract inner disk area from outer disk area before integrating. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The region is revolved about the x-axis.
- The slicing variable is x.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Build a volume from nested circular slices
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Rotate the region between y=x and y=x² for 0≤x≤1 about the x-axis. Find its volume.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
On [0,1], outer radius is x and inner radius is x²
The larger y value determines the outer boundary of each washer.
Work through the mathematics
Slice area=π(x²−x⁴)
Square each radius before subtracting their circular areas.
Check the conclusion
Volume=π∫₀¹(x²−x⁴)dx=2π/15
The antiderivatives give π(1/3−1/5).
The result
Volume=π∫₀¹(x²−x⁴)dx=2π/15
The antiderivatives give π(1/3−1/5).
Common mistakes to catch
- The outer curve can depend on the interval, so compare functions first.
- Rotation volume uses squared radii and cubic units.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Why is π(x−x²)² incorrect?
Show a hint
It squares the difference of radii.
Reveal answer and explanation
A washer is a difference of two disk areas
Its hole must be removed after computing each disk area.
Practice 2
Where does the washer hole close?
Show a hint
Set the inner radius x² to zero.
Reveal answer and explanation
At x=0
Away from zero the inner radius is positive.
Take the idea with you
Choose slices perpendicular to an axis and identify both radii before integrating.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Analyze repeated measurements through within-pair changes
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