How does speed give parametric arc length?
You will be able to: Derive and evaluate an integral of the magnitude of the coordinate rates.
How does speed give parametric arc length?
A small move on a map has both an east-west change and a north-south change. Pythagoras combines them into the length of that move.
A useful starting point: Why must we divide by x′ again for concavity? →
Words and symbols before equations
- Speed
- √((x′)²+(y′)²), when the parameter is time.
- Arc length
- Accumulated path length.
- Smooth parameterization
- Here x and y have continuous first derivatives.
- Traversal
- One passage through a portion of a curve.
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
A short chord has length √((Δx)²+(Δy)²). For positive Δt, rewrite this as √((Δx/Δt)²+(Δy/Δt)²)Δt.
Taking the sum limit gives L=∫ₐᵇ√((x′)²+(y′)²)dt. Both coordinate rates are squared; negative components cannot cancel distance.
If the path is traversed once, the integral gives its geometric length. Retraced portions count again. If a symbolic antiderivative is inconvenient, preserve the correct integral and evaluate numerically.
A worked example, step by step
A path is x=3t, y=4t for 0≤t≤2 seconds. Find its length.
- x′=3 m/s and y′=4 m/s.
- The speed is √(9+16)=5 m/s.
- L=∫₀²5dt=10 m.
- The endpoints (0,0) and (6,8) are 10 m apart; the straight path confirms the result.
Adding x′ and y′ does not give speed. The magnitude must be taken before integration.
Why square both coordinate rates?
Compare with an explanation
They are perpendicular components; Pythagoras combines their magnitudes.
Predict. Change one thing. Explain.
For a radius-2 circle, increase the number of quarter turns. Compare the distance 2t with the straight-line endpoint separation. Explain the full-turn case.
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=5 cos t, y=5 sin t on [0,π], find speed and path length, then compare with endpoint separation.
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Compare with the answer and four-point rubric
- 1 point: x′=−5 sin t and y′=5 cos t.
- 1 point: Speed=√(25 sin²t+25 cos²t)=5.
- 1 point: Length=5π units.
- 1 point: Endpoints (5,0) and (−5,0) are 10 units apart, less than 5π.
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 integrated for length?
The magnitude of the coordinate-rate vector.
RECALL 2What units does length have?
Linear coordinate units.
RECALL 3Is endpoint separation generally arc length?
No; the curved route may be longer.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How does speed give parametric arc length?
- L=∫ₐᵇ√((dx/dt)²+(dy/dt)²)dt, a<b.
- For time t, integrand units are length/time.
- Check whether the parameter interval repeats the path.
Remember: Adding x′ and y′ does not give speed. The magnitude must be taken before integration.
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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