How do you choose dx or dy for a length integral?
You will be able to: Set up and evaluate graph arc length using a consistent variable, bounds and derivative.
How do you choose dx or dy for a length integral?
A designer describes one edge as height in terms of horizontal position, but another as horizontal position in terms of height. Either description can measure the same physical kind of length.
A useful starting point: How do short straight pieces measure a curved path? →
Words and symbols before equations
- y=f(x)
- Height described as a function of horizontal position.
- x=g(y)
- Horizontal position described as a function of height.
- Integration bounds
- Endpoints measured in the chosen variable.
- Numerical integral
- A calculator approximation when an antiderivative is inconvenient.
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.
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.
- L = ∫ from 0 to 3 of √(1+x²/4) dx = 3.89893 m. Both reflected descriptions have equal length.
- 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 y=f(x), differentiate with respect to x and use x bounds: L=∫ₐᵇ √(1+[f′(x)]²)dx. For x=g(y), differentiate with respect to y and use y bounds: L=∫꜀ᵈ √(1+[g′(y)]²)dy.
Choose a variable for which the curve is a single-valued smooth graph on the interval. A vertical tangent in the x description can make the y description simpler. If one representation fails, do not blindly apply its smooth-graph formula.
For y=x²/4 from x=0 to 4, f′=x/2 and L=∫₀⁴ √(1+x²/4)dx≈5.916 m. This numerical answer is length; keep the exact integral setup and round only at the end.
Squaring a derivative removes its sign, but does not remove the square root or permit splitting √(1+u²) into 1+u. For a suitable perfect square, exact evaluation can be simple.
A worked example, step by step
For x=(2/3)y^(3/2) from y=0 to y=3 meters, find the curve length. Treat this as a mathematical model with coordinates measured in meters.
- Use y as the integration variable, with bounds 0 and 3.
- dx/dy=√y, which is continuous on [0,3].
- L=∫₀³ √(1+y)dy=[(2/3)(1+y)^(3/2)]₀³.
- L=(2/3)(8−1)=14/3 m. The bounds belong to y, not to x.
Do not mix dx/dy with dx or use x-coordinate bounds in a dy integral. Square the entire derivative before adding 1.
If x=g(y), what derivative belongs under the radical?
Compare with an explanation
dx/dy=g′(y), with dy and y-coordinate bounds.
Predict. Change one thing. Explain.
Switch between y=x²/4 and x=y²/4, then move the endpoint. The pictures are reflections across y=x. Explain why equal endpoint parameters give equal lengths even though the integration variable changes.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
L = ∫ from 0 to 3 of √(1+x²/4) dx = 3.89893 m. Both reflected descriptions have equal length.
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.
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 questionFor y=(2/3)x^(3/2) on [0,8], set up and evaluate its length. Explain why ∫₀⁸√x dx is a different quantity.
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Compare with the answer and four-point rubric
- 1 point: dy/dx=√x, continuous on [0,8].
- 1 point: L=∫₀⁸ √(1+x)dx.
- 1 point: L=[(2/3)(1+x)^(3/2)]₀⁸=(2/3)(27−1)=52/3 length units.
- 1 point: ∫₀⁸√x dx integrates the derivative and gives vertical change, omitting horizontal travel.
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 1Which bounds go with dy?
The starting and ending y coordinates, ordered increasingly for this length formula.
RECALL 2Can numerical evaluation be legitimate?
Yes. State the exact integral, then the appropriately rounded approximation.
RECALL 3Can √(1+u²) be replaced by 1+u?
No. The square root does not distribute over addition.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you choose dx or dy for a length integral?
- y=f(x): L=∫ √(1+[dy/dx]²)dx over x bounds.
- x=g(y): L=∫ √(1+[dx/dy]²)dy over y bounds.
- State the exact setup before a numerical approximation.
Remember: Do not mix dx/dy with dx or use x-coordinate bounds in a dy integral. Square the entire derivative before adding 1.
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.
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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