How do disk slices change around the y-axis?
You will be able to: Express radius as a function of y and use y-bounds.
How do disk slices change around the y-axis?
Stack round coasters upward to make a tower. Their radii may vary with height, so each horizontal slice uses the height variable y.
A useful starting point: How does rotating a filled strip create a disk? →
Words and symbols before equations
- Horizontal slice
- A section perpendicular to the vertical y-axis.
- R(y)
- Radius written using y.
- dy
- Small thickness along the y-axis.
- Inverse description
- Rewriting a boundary to give x in terms of y.
What this picture assumes
Original model; numerical labels are rounded. Disks 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=0, area π(R²−r²)=6.28319 square units. V=25.1327 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 0≤x≤√y for 0≤y≤4 around the y-axis. A horizontal segment reaches the axis and sweeps a disk of radius √y.
A(y)=π(√y)²=πy, so V=π∫₀⁴y dy=8π.
The boundary y=x² uses the nonnegative branch x=√y in this region. A different region might require another branch; the geometry decides.
Changing the axis changes the slice direction. All radii, bounds and the differential must use the same integration variable.
A worked example, step by step
Rotate the region 0≤x≤y² on 0≤y≤2 about the y-axis.
- Use horizontal slices perpendicular to y.
- R(y)=y².
- V=π∫₀²(y²)²dy=π∫₀²y⁴dy.
- Volume=32π/5 cubic units.
Do not insert y=x² directly as a radius when stacking along y. Solve for the needed horizontal distance in terms of y.
If R=√y, is disk area π√y?
Compare with an explanation
No. Squaring the radius gives πy.
Predict. Change one thing. Explain.
Move the horizontal disk between y=0 and y=4. Explain how the generating width √y becomes its radius and why the area grows linearly in y.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At y=2: R=1.41421, r=0, area π(R²−r²)=6.28319 square units. V=25.1327 cubic units over [0,4]. Radii are measured from x=0.
Original model; numerical labels are rounded. Disks 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≤2y on 0≤y≤1 around the y-axis. Set up and evaluate the volume integral.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Horizontal slice radius is R=2y.
- 1 point: Area is π(2y)²=4πy².
- 1 point: V=∫₀¹4πy²dy.
- 1 point: Volume=4π/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 determines the differential?
The direction in which perpendicular sections are stacked.
RECALL 2How do you choose a square-root branch?
Use the side of the axis occupied by the specified region.
RECALL 3Which bounds accompany dy?
y coordinates.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do disk slices change around the y-axis?
- Vertical axis → horizontal slices → dy.
- A(y)=πR(y)².
- Use bottom/top y-bounds.
Remember: Do not insert y=x² directly as a radius when stacking along y. Solve for the needed horizontal distance in terms of y.
Conditions: Original model; numerical labels are rounded. Disks 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 8.9, 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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