Finding the Area Between Curves Expressed as Functions of y
Choose horizontal strips with correct branches, right-minus-left boundaries, and matching y-bounds.
College Board Topic 8.5 · CED p. 151 (PDF p. 156) · CED effective Fall 2020; project course outline 2026–27.
A horizontal strip is just as valid. Write the boundary curves as x=R(y) and x=L(y); area is ∫[R(y)−L(y)]dy over the appropriate y-bounds. Right minus left replaces top minus bottom. The integration variable must match the strip width dy.
For x=y and x=y² on 0≤y≤1, the line is right of the parabola, so the area is ∫₀¹(y−y²)dy=1/6. Choosing y can avoid splitting a region that would require several x-integrals. When solving a curve for x or y, keep the correct branch; a square root may have both signs and only one may belong to your region.
Let's work through it
- For the region 0≤y≤2 between x=0 and x=y², the right boundary is y². Reason: Read the horizontal position of both boundaries.
- Use ∫₀² y²dy=[y³/3]₀²=8/3. Reason: Right-minus-left width is integrated with respect to y.
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 area integrand for horizontal strips?
See every recall question with its answer
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Retrieval 1
What is the area integrand for horizontal strips?
Worked answer and reason
Right boundary minus left boundary.
The strip’s horizontal length must be nonnegative.
Retrieval 2
What limits go with a dy integral?
Worked answer and reason
y-values.
The bounds describe the independent integration variable.
Catch the tempting mistake
- The formulas top-minus-bottom and right-minus-left are interchangeable without changing variables.
- They describe different strip directions. Choose a direction, express both boundaries accordingly, and use matching limits.
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 11 · No calculator · analytic
- Unit 8 problem 12 · No calculator · analytic
For the teacher
- Present a sideways parabola. Ask students to compare how many pieces each strip direction needs, then justify the simpler choice.
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.