Predict what a rotating region fills

Try

Place a filled rectangle next to a vertical axis and rotate it through a full turn. Each point traces a circle whose radius is its perpendicular distance from the axis. A horizontal interval of the rectangle therefore sweeps a disk if it reaches the axis, or an annulus if it starts away from it.

The filled generating region matters. Rotating only its far vertical edge produces a cylindrical surface; rotating the entire rectangle fills the interior. Do not use the same word cylinder to hide the distinction between surface and solid when calculating volume.

Predict

Predict the difference between rectangles 0≤x≤3, 0≤z≤5 and 1≤x≤3, 0≤z≤5, each rotated about the z-axis. Their outer profiles match, but one leaves a central hole. Then predict what happens to the first rectangle if the axis changes to the horizontal x-axis. The perpendicular radius and axial length exchange roles.

Experiment

One rectangle, different rotation axes

Rotate the entire filled rectangle through one full turn about the dashed line in its plane. Predict the solid and the area of a section perpendicular to its axis. Rotation takes the union of swept points: overlapping sweeps do not count as extra material.

Rectangle width12345 Rectangle height12345 Axis directionVerticalHorizontal Axis placementOn the left or bottom edgeThrough the middleOne unit beyond the left or bottom edge Reset rotation

Predict before revealing

Does the axis pass through the region, touch its boundary, or miss it? Identify the nearest and farthest distances from the axis. Decide whether the perpendicular section has a hole.

Reveal swept section

Try a fresh rectangle and sketch its predicted section before revealing. These controls model ideal filled regions and a full turn; wire outlines and partial turns are different problems. The experiment is temporary and awards no mastery.

The live rectangle settings above are temporary experiments. The fixed diagrams below are separate worked examples; their dimensions do not change with the controls.

Use these equal-scale generating-region drawings. Trace a horizontal segment at z=2 and list all its distances from the axis before imagining the turn. The first range is 0 through 3; the second is 1 through 3.

Draw the two horizontal section shapes on another sheet at the same scale. Their areas are 9π and 8π square units. Once the cylinder volume rule is justified, the volumes are 45π and 40π cubic units. The removed central tube is real missing volume, not a drawing artifact.

Understand

A triangular generating region produces a cone when one leg lies on the rotation axis. Suppose the vertices in the x,z plane are (0,0),(3,0),(0,6), with rotation about z. Similar triangles give the horizontal radius r(z)=3(1−z/6) for 0≤z≤6. The radius decreases linearly from 3 to zero; the disk area πr(z)² does not decrease linearly. That distinction explains why averaging just the top and bottom areas cannot determine the volume.

A filled right half-disk x≥0, x²+z²≤R² generates a solid ball when rotated about its vertical diameter. At each height, the interval from zero to √(R²−z²) sweeps exactly the sphere section disk. The circular arc alone generates the spherical surface instead. The section rule proves the identification without needing a volume formula first.

Rotation need not preserve a planar region’s area as a useful predictor of volume. For the first rectangle, rotating about x creates radius 5 and axial length 3 rather than radius 3 and length 5. The volumes are therefore different. Also beware of adding overlapping sweeps: either half of a full disk already generates the same ball around a diameter. Adding their swept volumes would double-count the interior.

Apply

For each practice case, name the axis, input region and height coordinate. At a fixed height, give the interval of distances to the axis; this determines whether the section is a disk, an annulus or empty. Sketch a few sections, but use the general distance rule to justify the complete solid.

Volume calculations in this lesson can be revisited after the next formula derivations. The shape identification itself does not depend on accepting the volume formulas from memory.

Axis-change transfer with a retained hole

Use the filled rectangle 0≤x≤2, 1≤y≤3. About the x-axis, every perpendicular section sweeps radii from 1 to 3. The section is an annulus, with area π(3²−1²)=8π. Its axial extent is two units, so the volume is 16π cubic units. The area of the original rectangle is not the rotating section area, and the missing central tube must not be filled in by the sketch.

Rotate the same rectangle about the y-axis instead. Now x measures radial distance and y measures axial position. The radial interval is [0,2], so the section is a filled disk of area 4π; the axial length is 3−1=2. The volume is 8π. The unchanged generating region does not imply unchanged volume because the rotation axis is part of the model.

The experiment replaces the upper y-boundary 3 by R while keeping the other boundaries fixed. Predict each volume before changing R. About x it is 2π(R²−1); about y it is 4π(R−1). At R=1 the region collapses to a line segment and both three-dimensional volumes are zero. This boundary case checks that the formulas reflect the actual region rather than a memorized outer profile.

For each retry, sketch a section perpendicular to the stated axis, label inner and outer radii, and write area times axial length with units. A drawing of the outer surface alone omits the evidence needed to decide whether the solid contains a hole.

Change the axis, rebuild the sections

Review criterion: Use perpendicular distances to the chosen axis and distinguish a filled solid from its boundary.

Predict, then change one input

Outer y-coordinate R (length units):

3

x-axis section area (square units)

25.132741

x-axis volume (cubic units)

50.265482

y-axis volume (cubic units)

25.132741

Region: 0≤x≤2, 1≤y≤R. About x the sections are annuli; about y they are radius-2 disks with height R−1. Values containing π are displayed approximately.

Explain the changed case: A hole survives

Rotate the filled rectangle 0≤x≤2, 1≤y≤3 about the x-axis. What is the volume?

• Axial length: 2
• Perpendicular distances from the x-axis: 1 through 3

Write a reason before selecting a claim. Feedback is public practice, not a proof grader.

18π cubic units16π cubic units8π cubic units

Switch to changed retryReset this practice

Inputs and choices are temporary. Nothing is saved, no lesson completion or score is changed, and repeating a public case is not unseen assessment.

Continue

The next lesson explains the circle measures needed for those disk and annulus areas. We will bound circles with polygons and make the approximation error shrink, instead of declaring that a finite many-sided figure is already a circle.