How does rotating a filled strip create a disk?
You will be able to: Build a disk volume integral perpendicular to the x-axis.
How does rotating a filled strip create a disk?
Rotate a filled triangle around one edge and it sweeps out a cone. Each thin strip touching that edge sweeps a full circular disk with no central hole.
A useful starting point: Why must a diameter be halved before calculating area? →
Words and symbols before equations
- Axis of revolution
- The fixed line around which a region rotates.
- Disk radius R
- Perpendicular distance from the axis to the outer boundary.
- Slice thickness dx
- Thickness along the x-axis.
- Solid of revolution
- The filled volume swept out by rotating the region.
What this picture assumes
Original model; numerical labels are rounded. Disks about the x-axis. Vertical generating segments rotate around y=0; 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=2, r=0, area π(R²−r²)=12.5664 square units. V=27.2271 cubic units over [0,2]. Radii are measured from y=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
For 0≤x≤2, rotate the region 0≤y≤1+x around the x-axis. Each vertical segment starts on the axis, so its rotation has no hole and radius R=1+x.
Cross-sectional area is πR²=π(1+x)². Volume is π∫₀²(1+x)²dx=26π/3 cubic units.
The volume integral adds circular areas times thickness along the axis. A vertical slice is perpendicular to the horizontal axis, so dx is the correct thickness.
Rotate the filled region, not just its boundary curve. The curve creates a surface; the full region supplies the filled disks inside.
A worked example, step by step
Rotate 0≤y≤√x on [0,4] about the x-axis. Find volume.
- The region reaches the axis, so slices are disks.
- R=√x, hence A=πx.
- V=π∫₀⁴ x dx.
- The volume is 8π cubic units.
The radius is squared before integration: ∫πR²dx is not π(∫R dx)².
Is rotating only the boundary curve enough to specify the filled volume?
Compare with an explanation
It traces the surface; the rotated filled region gives the volume inside.
Predict. Change one thing. Explain.
Move the disk along the x-axis. Compare radius 1+x, disk area and the generating vertical segment. Rotate the view to see that the central hole is absent.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At x=1: R=2, r=0, area π(R²−r²)=12.5664 square units. V=27.2271 cubic units over [0,2]. Radii are measured from y=0.
Original model; numerical labels are rounded. Disks about the x-axis. Vertical generating segments rotate around y=0; 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 the region 0≤y≤x for 0≤x≤3 about the x-axis. Find volume and check against cone geometry.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Radius is x.
- 1 point: V=π∫₀³x²dx.
- 1 point: Volume=9π.
- 1 point: A cone with radius 3 and height 3 has (1/3)π×9×3=9π, confirming the result.
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 1When is a section a disk?
The filled radial segment reaches the axis.
RECALL 2What is squared?
The radius at each slice.
RECALL 3What are volume units?
Cubic coordinate units.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How does rotating a filled strip create a disk?
- Disk A=πR².
- Horizontal axis → perpendicular vertical slices → dx.
- Volume=∫slice area×thickness.
Remember: The radius is squared before integration: ∫πR²dx is not π(∫R dx)².
Conditions: Original model; numerical labels are rounded. Disks about the x-axis. Vertical generating segments rotate around y=0; 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 BC 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 BC topics 8.1–8.13, including smooth planar graph arc length and distance traveled. Focused lesson titles, examples, questions and illustrations are original Refresh Kid material. Unit 7 differential equations is available separately. Parametric and polar curves belong to later units.
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 and Arc Length Calculus Problems video titles and creator 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 and the arc-length subsection of Volume 2 Section 2.4 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 BC 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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