How do we identify a washer’s horizontal radii?
You will be able to: Use right and left boundaries as distances from the y-axis with dy.
How do we identify a washer’s horizontal radii?
A hollow tower can widen with height. A horizontal slice shows an outer rim and an inner opening, both measured from the same vertical centerline.
A useful starting point: Why is washer area the difference of two squares? →
Words and symbols before equations
- Horizontal radius
- An x-distance measured at fixed y.
- Near/far boundary
- The boundary nearer/farther from the axis.
- y-limits
- Heights of the first and last slices.
- Domain branch
- The part of a curve bounding the specified region.
What this picture assumes
Original model; numerical labels are rounded. Washers about the y-axis. Horizontal generating segments rotate around x=0; sections stack using dy over [0,4]. 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 y=2: R=1.41421, r=1, area π(R²−r²)=3.14159 square units. V=8.37758 cubic units over [0,4]. Radii are measured from x=0.
- 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 y/2≤x≤√y for 0≤y≤4 around the y-axis. Both boundaries are nonnegative, and √y≥y/2 on this interval.
R=√y and r=y/2, so A=π(y−y²/4). Thus V=π∫₀⁴(y−y²/4)dy=8π/3.
The intersections solve √y=y/2, giving y=0,4 after checking both roots in the original equation. These heights supply the bounds.
The right boundary is farther from the y-axis here because the whole region is on its right. For a region on the left, coordinate order alone would not identify the outer radius; distances must be checked.
A worked example, step by step
Rotate 1≤x≤y on 1≤y≤3 about the y-axis.
- Use horizontal slices with y bounds 1 and 3.
- Outer radius is y and inner radius 1.
- V=π∫₁³(y²−1)dy.
- Evaluate π[y³/3−y]₁³=20π/3 cubic units.
Rightmost does not always mean outermost for a vertical axis. It does in this positive-side example; always compare distances to the actual axis.
Can the area vanish when both radii are nonzero?
Compare with an explanation
Yes. At y=4 both radii equal 2, so their squared areas cancel.
Predict. Change one thing. Explain.
Move the horizontal slice from y=0 to y=4. Compare √y and y/2 and explain why the washer closes at both ends of the region.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At y=2: R=1.41421, r=1, area π(R²−r²)=3.14159 square units. V=8.37758 cubic units over [0,4]. Radii are measured from x=0.
Original model; numerical labels are rounded. Washers about the y-axis. Horizontal generating segments rotate around x=0; sections stack using dy over [0,4]. 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≤x≤2 and 1≤y≤3 about the y-axis, then remove the core 0≤x<1. Find the remaining volume.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: The resulting region has 1≤x≤2 over y from 1 to 3.
- 1 point: R=2 and r=1.
- 1 point: V=π∫₁³(4−1)dy.
- 1 point: The volume is 6π 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 is a horizontal radius?
An x-distance from the vertical rotation line.
RECALL 2Why must both boundary branches be checked?
Only the branches describing the actual region give valid radii.
RECALL 3What happens where R=r?
The section has zero area.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do we identify a washer’s horizontal radii?
- Vertical axis → washers stacked using dy.
- R and r are horizontal distances.
- V=π∫(R(y)²−r(y)²)dy.
Remember: Rightmost does not always mean outermost for a vertical axis. It does in this positive-side example; always compare distances to the actual axis.
Conditions: Original model; numerical labels are rounded. Washers about the y-axis. Horizontal generating segments rotate around x=0; sections stack using dy over [0,4]. 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 8.11, 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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