How do both radii change around a shifted horizontal line?
You will be able to: Measure both washer radii from an offset axis and recognize axes above a region.
How do both radii change around a shifted horizontal line?
Move a washer’s axle below its material strip. Both the near and far edges become farther from the axle, even if the strip itself keeps the same thickness.
A useful starting point: How do we identify a washer’s horizontal radii? →
Words and symbols before equations
- Axis offset k
- The y-coordinate of a horizontal rotation line.
- Distance to a line
- abs(y−k).
- Outer/inner assignment
- Based on larger/smaller distance, not function name.
- Crossing an axis
- A situation that can remove the hole and require a different setup.
What this picture assumes
Original model; numerical labels are rounded. Washers about y=−1. Vertical generating segments rotate around y=-1; sections stack using dx over [0,2]. Radii are distances from that axis. Curved mesh is sampled for display; formulas determine area and volume.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- At x=1: R=3, r=1, area π(R²−r²)=25.1327 square units. V=52.3599 cubic units over [0,2]. Radii are measured from y=-1.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Rotate 0≤y≤1+x on [0,2] around y=−1. The axis lies below the region, so R=(1+x)−(−1)=2+x and r=0−(−1)=1.
V=π∫₀²[(2+x)²−1]dx=50π/3. The gap below the region creates a hole, even though the original region touched the x-axis.
If the axis lies above a region, the lower boundary can be farther away. For −1≤y≤1 rotated about y=3, R=4 and r=2.
If a filled slice straddles the rotation axis, the swept section is a disk with radius the farther endpoint distance. Do not subtract a fictitious inner disk; split into cases if the geometry changes along the interval.
A worked example, step by step
Rotate the region 1≤y≤2 on 0≤x≤3 about y=4.
- The axis is above both boundaries.
- Distance to lower boundary 1 is R=3; distance to upper boundary 2 is r=2.
- V=π∫₀³(9−4)dx.
- Volume=15π cubic units; the lower curve supplies the outer radius here.
Do not automatically call the upper curve the outer radius. Measure both distances from the actual axis and inspect whether the slice crosses it.
If a slice runs from y=−1 to y=2 and rotates about y=0, is the inner radius 1?
Compare with an explanation
No. The filled slice crosses the axis and sweeps a disk of radius 2.
Predict. Change one thing. Explain.
Inspect the orange axis y=−1 and both radii at a movable x. Compare this solid with rotation about y=0 and explain why shifting only the axis increases volume.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At x=1: R=3, r=1, area π(R²−r²)=25.1327 square units. V=52.3599 cubic units over [0,2]. Radii are measured from y=-1.
Original model; numerical labels are rounded. Washers about y=−1. Vertical generating segments rotate around y=-1; sections stack using dx over [0,2]. Radii are distances from that axis. Curved mesh is sampled for display; formulas determine area and volume.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the units, signed change, strip direction, radius distances or cross-sectional area. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionRotate 0≤y≤x on 0≤x≤2 about y=−2. Find both radii and the volume.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: R=x+2 and r=2.
- 1 point: A=π[(x+2)²−4]=π(x²+4x).
- 1 point: V=π∫₀²(x²+4x)dx.
- 1 point: Volume=π(8/3+8)=32π/3 cubic units.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What creates an inner hole?
A gap between the filled slice and the rotation axis.
RECALL 2What determines outer radius?
The farther perpendicular distance from the axis.
RECALL 3What if the axis cuts through the slice?
Use the full swept disk and reassess any interval changes.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do both radii change around a shifted horizontal line?
- R=farther distance to axis; r=nearer distance if a gap exists.
- A=π(R²−r²).
- A slice crossing the axis has no inner hole.
Remember: Do not automatically call the upper curve the outer radius. Measure both distances from the actual axis and inspect whether the slice crosses it.
Conditions: Original model; numerical labels are rounded. Washers about y=−1. Vertical generating segments rotate around y=-1; sections stack using dx over [0,2]. Radii are distances from that axis. Curved mesh is sampled for display; formulas determine area and volume.
Refresh Kid · AP Calculus AB Unit 8 · Objectives CHA-5.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.12, CHA-5.C. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. Unit 8 covers AB topics 8.1–8.12. Topic 8.13 (arc length) is BC-only. Focused lesson titles, examples, questions and illustrations are original Refresh Kid material. Unit 7 is a separate curriculum outline, not a prerequisite gate to these lessons.
Average value is distinguished from average rate. Motion uses velocity for displacement and speed for distance, with initial values stated separately. Area bounds and ordering are checked, including multiple crossings. Volumes derive the slice area before integration, distinguish diameter from radius, and use distances from the specified axis. Disks and washers use perpendicular slices and a consistent integration variable. A region crossing an axis requires checking the actual swept disk rather than inventing a hole.
The Organic Chemistry Tutor Disk & Washer Method video title, creator and relevant description were checked; the full video was not reviewed. Use the free video as an optional companion; no paid material is required. Khan Academy’s unit destination was checked, but its JavaScript lesson contents were not fully readable by the research tool. OpenStax Sections 6.1 and 6.2 were consulted for conceptual cross-checking. No provider scripts, questions, artwork or diagrams were copied. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional spatial inspection and camera controls. All cross-section and revolution geometry is original and uses existing self-hosted Three.js with its MIT license. Spatial coordinates preserve the mathematical lengths; sampled mesh surfaces illustrate exact formulas. Teal shows the solid and orange a selected zero-thickness section. The volume is for the entire solid, not the highlighted plane. No autoplay is used; camera rotation changes only the view. Labeled 2D regions, cross-section diagrams, readouts and calculations remain available without WebGL.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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