Why must we split some area integrals?
You will be able to: Locate all crossings and integrate nonnegative boundary differences piecewise.
Why must we split some area integrals?
Two trails weave across each other. One is above on the first stretch and below on the next. A single signed subtraction would cancel parts of the land between them.
A useful starting point: When are horizontal strips simpler? →
Words and symbols before equations
- Crossing
- An intersection at which boundary order changes.
- Piecewise integral
- A sum over intervals with a consistent formula.
- Signed difference
- f−g, which may change sign.
- Geometric area
- The integral of abs(f−g).
What this picture assumes
Original model; numerical labels are rounded. y=x³ and y=x intersect at −1,0,1. Upper boundary changes at zero. Two lobe areas add to 1/2; the unsplit signed difference cancels.
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.
- x=0.5; x³=0.125, x=0.5; positive strip height=0.375. Line is upper on (0,1). Lobe areas 1/4+1/4=1/2; signed difference integral=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
The curves y=x³ and y=x intersect at x=−1,0,1. On (−1,0), x³>x; on (0,1), x>x³.
The two enclosed lobes have total area ∫₋₁⁰(x³−x)dx+∫₀¹(x−x³)dx=1/4+1/4=1/2.
The unsplit integral ∫₋₁¹(x³−x)dx is zero because the difference is odd. Zero signed difference does not mean no region exists.
Find all intersections, test the ordering between adjacent ones, and add positive areas. An intersection without an order change may be used as a split but does not require switching subtraction.
A worked example, step by step
Find the area between y=sin x and y=0 on [0,2π].
- Zeros are 0,π,2π.
- Sine is positive on (0,π) and negative on (π,2π).
- Use the integral of sin x from 0 to π minus the integral of sin x from π to 2π.
- The two areas are 2 and 2, totaling 4; the signed integral alone is zero.
Do not take abs of the final integral to recover lost area after cancellation. Split first or integrate the absolute difference.
Does every intersection force a change of upper curve?
Compare with an explanation
No. Curves can touch without crossing; inspect the side signs.
Predict. Change one thing. Explain.
Move a vertical slice from −1 to 1. Identify which curve is upper on each side of zero and explain why the full signed integral cancels.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
x=0.5; x³=0.125, x=0.5; positive strip height=0.375. Line is upper on (0,1). Lobe areas 1/4+1/4=1/2; signed difference integral=0.
Original model; numerical labels are rounded. y=x³ and y=x intersect at −1,0,1. Upper boundary changes at zero. Two lobe areas add to 1/2; the unsplit signed difference cancels.
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 questionFind the area between y=x² and y=1 on [−2,2], identifying every needed split.
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Compare with the answer and four-point rubric
- 1 point: Intersections are −1 and 1.
- 1 point: x² is upper on [−2,−1] and [1,2]; 1 is upper on [−1,1].
- 1 point: Area=2∫₁²(x²−1)dx+∫₋₁¹(1−x²)dx.
- 1 point: This is 2(4/3)+4/3=4 square units.
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 1Why does a signed integral cancel?
Its difference changes sign.
RECALL 2What must be checked between crossings?
Which boundary is upper or right.
RECALL 3What function directly represents area height?
The absolute difference of the boundary functions.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Why must we split some area integrals?
- Area=∫abs(f−g)dx.
- Split at every ordering change.
- Sum nonnegative piece areas.
Remember: Do not take abs of the final integral to recover lost area after cancellation. Split first or integrate the absolute difference.
Conditions: Original model; numerical labels are rounded. y=x³ and y=x intersect at −1,0,1. Upper boundary changes at zero. Two lobe areas add to 1/2; the unsplit signed difference cancels.
Refresh Kid · AP Calculus AB Unit 8 · Objectives CHA-5.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.6, CHA-5.A. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. Unit 8 covers AB topics 8.1–8.12. Topic 8.13 (arc length) is BC-only. Focused lesson titles, examples, questions and illustrations are original Refresh Kid material. Unit 7 is a separate curriculum outline, not a prerequisite gate to these lessons.
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 video title, creator and relevant description 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 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 AB 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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