Finding the Area Between Curves That Intersect at More Than Two Points
Split at relevant crossings so positive area does not disappear through signed cancellation.
College Board Topic 8.6 · CED p. 152 (PDF p. 157) · CED effective Fall 2020; project course outline 2026–27.
When the upper curve changes, split the interval at every relevant intersection and integrate a positive difference on each piece. Equivalently, area is ∫|f−g|. The integral ∫(f−g) is signed and can cancel, so it may be much smaller than the area.
For y=x and y=x³ on [−1,1], intersections are −1,0,1. On [−1,0], x³ is above x; on [0,1], x is above x³. Therefore area is ∫₋₁⁰(x³−x)dx+∫₀¹(x−x³)dx=1/2. Symmetry can save work after the sign pattern is justified. A calculator graph may miss a tangency or a closely spaced crossing; solve or verify intersections carefully before relying on a plot.
Let's work through it
- For y=sin x and the x-axis on [0,2π], split at π. Reason: Sine changes sign there.
- Area is ∫₀^π sin x dx−∫_π^(2π) sin x dx=2+2=4. Reason: Use positive heights on each sign interval.
- The signed integral over the whole interval is 0. Reason: The two lobes cancel when signs are retained.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Why can ∫(f−g)=0 while the area is positive?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Why can ∫(f−g)=0 while the area is positive?
Worked answer and reason
Opposite signed pieces can cancel.
Geometric area uses |f−g|.
Retrieval 2
Must an intersection where order does not change require a sign reversal?
Worked answer and reason
No.
Test the order on both sides; a tangency may not switch the upper curve.
Catch the tempting mistake
- Only the first and last intersection matter.
- Interior intersections can change the strip height formula. Check every relevant crossing or use absolute value with a justified setup.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 8 problem 13 · No calculator · analytic
- Unit 8 problem 14 · No calculator · graphical
For the teacher
- Use an example with three intersections and one tangency example. Ask students to build a sign table before any antiderivative work.
Glossary and deeper reading · 1 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.