How do short straight pieces measure a curved path?
You will be able to: Derive the graph arc-length integral from the Pythagorean theorem.
How do short straight pieces measure a curved path?
A flexible strip follows a curved garden border. Measuring only the horizontal width misses its rises and falls. Short straight chords give a first estimate of the strip needed.
A useful starting point: How do shifted vertical axes change the setup? →
Words and symbols before equations
- Arc length
- Distance along a curve, in length units.
- Chord
- Straight segment joining two points on the curve.
- Delta, Δ
- Change in a quantity, such as Δx.
- Smooth graph
- Here, y=f(x) has a continuous derivative on the closed interval.
What this picture assumes
Both axes measure meters at the same visual scale. The model is y=x²/4 (or its reflected x=y²/4 graph), with a continuous derivative. The teal curve is sampled for display; the length readout uses the exact antiderivative, rounded for display. No timing or speed is specified. Refinement levels 0–6 give 1, 2, 4, 8, 16, 32 and 64 equal-input chords. Orange segments approximate length from below.
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.
- 2 chords total 5.84162 m; smooth length 5.91577 m; gap 0.0741522 m. Endpoint separation 5.65685 m.
- 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 a short piece, horizontal change Δx and vertical change Δy are the legs of a right triangle. Its chord length is √((Δx)²+(Δy)²). Both axes must use compatible physical length units.
For increasing x, factor out positive Δx: chord length = √(1+(Δy/Δx)²) Δx. The secant slope Δy/Δx approaches the derivative f′(x) as pieces become short.
Adding the chords and taking the limit gives L=∫ₐᵇ √(1+[f′(x)]²) dx for a<b and continuous f′. The 1 counts horizontal motion; the derivative term counts vertical motion.
A finite chord sum is an approximation and cannot exceed the length of this smooth curve. Refining a nested partition cannot decrease that sum. The integral gives length, not area under the graph.
A worked example, step by step
A straight ramp follows y=3x/4 from x=0 to x=4 meters. Find its length by geometry and by integration.
- The endpoints are (0,0) and (4,3), with both coordinates in meters.
- Pythagoras gives √(4²+3²)=5 m.
- Since f′=3/4, the length integrand is √(1+9/16)=5/4.
- L=∫₀⁴(5/4)dx=5 m, agreeing with geometry; horizontal width alone is 4 m.
Integrating f gives signed area, and integrating f′ gives net vertical change. Neither measures curved-path length.
Why does the integrand contain 1?
Compare with an explanation
Factoring Δx from the horizontal leg produces 1; omitting it ignores horizontal travel.
Predict. Change one thing. Explain.
For y=x²/4 on 0≤x≤4 meters, increase the number of straight pieces through 1, 2, 4, 8 and 16. Compare the orange chord total with the curved length. Explain why the endpoint distance remains fixed.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
2 chords total 5.84162 m; smooth length 5.91577 m; gap 0.0741522 m. Endpoint separation 5.65685 m.
Connect geometry and the integral: Each short chord has length √((Δx)²+(Δy)²). Summing and taking the limit gives the length integral. All lengths and comparisons are stated above.
Both axes measure meters at the same visual scale. The model is y=x²/4 (or its reflected x=y²/4 graph), with a continuous derivative. The teal curve is sampled for display; the length readout uses the exact antiderivative, rounded for display. No timing or speed is specified. Refinement levels 0–6 give 1, 2, 4, 8, 16, 32 and 64 equal-input chords. Orange segments approximate length from below.
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 questionA path y=−5x/12 runs from x=0 to x=12 meters. Set up its length integral, evaluate it, and compare length with vertical change.
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Compare with the answer and four-point rubric
- 1 point: The derivative is −5/12.
- 1 point: L=∫₀¹² √(1+25/144)dx.
- 1 point: The integrand is 13/12, so L=13 m.
- 1 point: Vertical change is −5 m; signed change differs from nonnegative path length.
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 replaces the secant slope in the limit?
The derivative f′(x).
RECALL 2When is this formula guaranteed here?
For a graph with continuous derivative on a finite closed interval.
RECALL 3Why is a chord estimate short?
A straight segment is no longer than the curved path joining its endpoints.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do short straight pieces measure a curved path?
- L=∫ₐᵇ √(1+[f′(x)]²)dx for a<b and continuous f′.
- Chord length=√((Δx)²+(Δy)²).
- Length has linear units; area has square units.
Remember: Integrating f gives signed area, and integrating f′ gives net vertical change. Neither measures curved-path length.
Conditions: Both axes measure meters at the same visual scale. The model is y=x²/4 (or its reflected x=y²/4 graph), with a continuous derivative. The teal curve is sampled for display; the length readout uses the exact antiderivative, rounded for display. No timing or speed is specified. Refinement levels 0–6 give 1, 2, 4, 8, 16, 32 and 64 equal-input chords. Orange segments approximate length from below.
Refresh Kid · AP Calculus BC Unit 8 · Objectives CHA-6.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.13, CHA-6.A. 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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