Finding the Area Between Curves Expressed as Functions of x
Find intersections and integrate a justified top-minus-bottom vertical strip.
College Board Topic 8.4 · CED p. 150 (PDF p. 155) · CED effective Fall 2020; project course outline 2026–27.
A vertical strip between y=f(x) and y=g(x) has height top minus bottom. If f≥g on [a,b], its area is ∫ₐᵇ[f(x)−g(x)]dx. Find intersections for the bounds, then test which function is above inside each interval. Area is nonnegative; a negative result usually signals reversed order or a missed crossing.
For y=x and y=x², intersections are x=0 and x=1, and x≥x² between them. The area is ∫₀¹(x−x²)dx=1/6. A graph is a useful guide, but verify bounds and ordering analytically or numerically when needed. A drawing that looks right does not establish the exact intersection value.
Let's work through it
- Solve x=x² to find 0 and 1. Reason: The curves’ intersections bound this enclosed region.
- Use top x minus bottom x². Reason: For 0<x<1, x>x².
- Evaluate [x²/2−x³/3]₀¹=1/6. Reason: The strip height integrates to positive area.
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
What is the strip height if the upper curve is 5 and lower is x²?
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
What is the strip height if the upper curve is 5 and lower is x²?
Worked answer and reason
5−x².
Measure vertical separation as top minus bottom.
Retrieval 2
If the curves cross inside the bounds, can you keep one order throughout?
Worked answer and reason
No, unless using absolute value.
The top curve changes at a crossing.
Catch the tempting mistake
- Area between curves equals the difference of their endpoint heights.
- Endpoint heights are not area. Integrate the vertical separation over the full interval.
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
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Put it to work
- Unit 8 problem 9 · No calculator · analytic
- Unit 8 problem 10 · No calculator · analytic
- Unit 8 problem 24 · Graphing calculator required · multiple
For the teacher
- Ask students to sketch one representative strip, label its height and width, and explain the integrand before evaluating.
Glossary and deeper reading · 2 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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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.