Can the same curve produce different distance integrals?
You will be able to: Distinguish a geometric curve from traversal count and parameter speed.
Can the same curve produce different distance integrals?
Two runners use the same circular track. One completes one lap and the other two. Their route drawings match, but their distance totals do not.
A useful starting point: How does speed give parametric arc length? →
Words and symbols before equations
- Geometric trace
- The set of points visited.
- Reparameterization
- A different rule for traversing the same path.
- Repeated traversal
- Passing over points again.
- Displacement
- Final position minus initial position.
What this picture assumes
Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. Radius-2 circle, angular rate 1 rad/s. Distance counts repeated traversals. The projection is the physical planar route; optional time height is not a third spatial coordinate.
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.
- t=1.5708 s; point (0, 2) m; V=⟨-2, 0⟩ m/s. Speed 2 m/s; distance 3.14159 m; endpoint separation 2.82843 m. Tangent: horizontal. Increasing t is counterclockwise.
- 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 x=2 cos t, y=2 sin t, speed is 2. The interval [0,2π] gives length 4π, while [0,4π] gives 8π.
For x=2 cos(2t), y=2 sin(2t), speed is 4. One lap now takes [0,π], still giving length 4π. Faster traversal does not change a single lap’s geometry.
Always inspect the interval and orientation before calling an integral the length of the geometric trace. A closed trip may have zero displacement and positive distance.
A worked example, step by step
Compare x=3 cos t, y=3 sin t on [0,2π] with x=3 cos(2t), y=3 sin(2t) on [0,2π].
- The first motion has speed 3.
- Its angle increases by 2π, so one lap has distance 6π.
- The second motion has speed 6 and its angle increases by 4π.
- It makes two laps and travels 12π; both motions end where they started.
Do not assume one traversal merely because the Cartesian equation describes one circle.
Does doubling speed always double a single lap’s length?
Compare with an explanation
No. It halves the time needed; one lap’s distance is unchanged.
Predict. Change one thing. Explain.
Move from four to eight quarter turns in the circle model. Explain why the planar curve stays the same while its time-lift representation adds another turn.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
t=1.5708 s; point (0, 2) m; V=⟨-2, 0⟩ m/s. Speed 2 m/s; distance 3.14159 m; endpoint separation 2.82843 m. Tangent: horizontal. Increasing t is counterclockwise.
Read the representation: Teal shows a curve; orange marks the selected point, tangent, ray or swept region described above. Dashed segments are auxiliary comparisons. Read the model conditions and units before comparing lengths.
Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. Radius-2 circle, angular rate 1 rad/s. Distance counts repeated traversals. The projection is the physical planar route; optional time height is not a third spatial coordinate.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the units, coordinate rates, parameter direction, tracing count or radial boundaries. 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 questionFor x=2 cos(3t), y=2 sin(3t), find speed, a one-lap interval starting at 0, and distance on that interval.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Derivatives are −6 sin(3t) and 6 cos(3t).
- 1 point: Speed is 6.
- 1 point: One lap requires 3t=2π, so use [0,2π/3].
- 1 point: Distance is 6(2π/3)=4π, the circumference of radius 2.
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 distinguishes trace from traversal?
A trace is a set; a traversal records movement and repetition.
RECALL 2How do you detect repeated circular motion?
Track the total angle change.
RECALL 3Does reversing direction make distance negative?
No; speed is nonnegative.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Can the same curve produce different distance integrals?
- Count how much the angle or parameter advances.
- A closed trip can have zero displacement and positive distance.
- Compare matching single-traversal intervals.
Remember: Do not assume one traversal merely because the Cartesian equation describes one circle.
Conditions: Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. Radius-2 circle, angular rate 1 rad/s. Distance counts repeated traversals. The projection is the physical planar route; optional time height is not a third spatial coordinate.
Refresh Kid · AP Calculus BC Unit 9 · Objectives CHA-6.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 9.3, CHA-6.B. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. This unit covers BC topics 9.1–9.9. Parametric and vector motion is planar. Polar arc length, surface area and three-dimensional vector calculus are not assigned as required Unit 9 material.
Parametric slope and second derivatives retain their nonzero-denominator conditions. Speed is the magnitude of velocity; displacement and distance use different integrals. Polar coordinates allow signed radii, with distance abs(r). Polar tangents use Cartesian angular rates. Areas use squared radial boundaries with correct angular intervals, tracing counts and boundary switches; the pole is checked separately.
All focused explanations, examples, practice and models are original Refresh Kid work. OpenStax was consulted for mathematical cross-checking. Khan Academy’s destination was checked, but JavaScript lesson content was not fully readable by the research tool. Organic Chemistry Tutor video titles, creator and destinations were checked; full videos were not reviewed. No questions, diagrams or provider scripts were copied. Resources are optional; Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional camera and spatial inspection. The original parameter-lift model uses self-hosted Three.js with its MIT license. The teal projection is the physical circle. The vertical axis of the orange lift is time, not spatial height, and its scale is explicitly stated. No spatial arc-length calculation is made from that lift. Keyboard controls and labeled 2D alternatives remain available, without autoplay or required 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 implementation has not been evaluated with learners.
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