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Grade 12 · Washer volumes

Washer volumes: A half-unit inner factor

Washer volumes: investigate a half-unit inner factor with outer radius factor a = 1.5; inner radius factor b = 0.5.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Washer volumes: A half-unit inner factor. Upper x bound: 0. Outer radius: 0. Cross-section area: 0. Accumulated volume: 0Subtract inner disk area from outer03012yx → · labeled axes rescale to this model
Rotation is about the horizontal axis, 0≤b<a throughout the permitted controls, and radii are nonnegative for x≥0. The drawing is a radius profile, not a perspective rendering; volume has cubic length units.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Upper x bound
0
Outer radius
0
Cross-section area
0
Accumulated volume
0

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From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

Rotating a region between two radius curves around an axis produces annular cross-sections. Each washer's area is the outer disk area minus the inner disk area. Squaring the difference of radii would incorrectly remove the cross term and describe a different area. This investigation starts with Outer radius factor a = 1.5; Inner radius factor b = 0.5. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

R=ax, r=bx; V(T)=π(a²−b²)T³/3

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Read the outer radius ax and inner radius bx at the moving x position. The starting case is “A half-unit inner factor.”

  2. STEP 2

    Follow the changing quantity

    Use π(R²−r²) for each cross-section while the accumulated interval grows from zero to two.

  3. STEP 3

    Explain and test the result

    Integrate the squared radii to obtain the cubic volume formula. Compare a solid cone with a hollow one using the same outer radius.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Outer radius factor a = 1.5; Inner radius factor b = 0.5. Pause the timeline at 60%. Given upper x bound = 1.2, calculate outer radius, cross-section area, accumulated volume. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

At x=1.2, π(R²−r²)=π[(1.5)²−(0.5)²](1.2)²=9.047787. Integrating from 0 to 1.2 gives π[1.5²−0.5²](1.2)³/3=3.619115. Results: Outer radius: 1.8; Cross-section area: 9.048; Accumulated volume: 3.619. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 12 · Calculus

Build a volume from nested circular slices

Subtract inner disk area from outer disk area before integrating.