How is radius measured from an axis that is not zero?
You will be able to: Measure distance from a shifted horizontal or vertical axis.
How is radius measured from an axis that is not zero?
A wheel’s radius is measured from its axle, even if the axle is above the floor. A rotation line y=1 is the reference line; the x-axis is no longer the center.
A useful starting point: How do disk slices change around the y-axis? →
Words and symbols before equations
- Shifted axis
- A line y=k or x=h.
- Perpendicular distance
- Nonnegative length from the axis to the boundary.
- Touching the axis
- A filled strip begins at the rotation line, leaving no hole.
- Translation
- Moving a shape without changing its size.
What this picture assumes
Original model; numerical labels are rounded. Disks 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=1, r=0, area π(R²−r²)=3.14159 square units. V=8.37758 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 the region 1≤y≤1+x on [0,2] about y=1. Each radius is (1+x)−1=x, not 1+x.
The section is a disk because the lower boundary is the axis itself. V=π∫₀²x²dx=8π/3.
For a vertical shifted line x=h, use horizontal slices and radius abs(g(y)−h). For example, rotating h≤x≤h+y on 0≤y≤2 gives the same volume 8π/3.
Translating both the region and its axis together changes neither radii nor volume. Translating only the axis relative to a fixed region can change the volume and introduce a hole.
A worked example, step by step
Rotate 2≤y≤2+√x on [0,4] about y=2.
- The filled region touches the axis at its lower boundary.
- Radius=(2+√x)−2=√x.
- A=πx and V=π∫₀⁴x dx.
- Volume=8π; the vertical shift cancels from the radius.
Use distance from the actual axis. The coordinate value y is a radius only when the reference axis is y=0 and the relevant distance equals that value.
Does raising both a region and its horizontal axis by 5 change its volume?
Compare with an explanation
No. All distances from the axis remain the same.
Predict. Change one thing. Explain.
Inspect the region from y=1 to y=1+x and the orange line y=1. Read the radius as a distance between them, then compare with the unshifted cone.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At x=1: R=1, r=0, area π(R²−r²)=3.14159 square units. V=8.37758 cubic units over [0,2]. Radii are measured from y=1.
Original model; numerical labels are rounded. Disks 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 3≤x≤3+y² on 0≤y≤2 around x=3. Explain the slice direction and find volume.
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Compare with the answer and four-point rubric
- 1 point: The axis is vertical, so use horizontal sections and dy.
- 1 point: R=(3+y²)−3=y².
- 1 point: V=π∫₀²y⁴dy.
- 1 point: Volume=32π/5 cubic units; there is no hole because the strip reaches x=3.
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 1Where is a radius measured from?
The specified rotation axis.
RECALL 2When does a shifted region still make disks?
When its filled strips reach the axis.
RECALL 3What does a common translation preserve?
Distances, section areas and volume.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How is radius measured from an axis that is not zero?
- Horizontal y=k: R=abs(boundary−k).
- Vertical x=h: R=abs(boundary−h).
- No gap to the axis ⇒ disk.
Remember: Use distance from the actual axis. The coordinate value y is a radius only when the reference axis is y=0 and the relevant distance equals that value.
Conditions: Original model; numerical labels are rounded. Disks 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 8.10, 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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