Volumes with Cross Sections: Squares and Rectangles
Turn a stated square or rectangular cross section into an area function before integrating volume.
College Board Topic 8.7 · CED p. 153 (PDF p. 158) · CED effective Fall 2020; project course outline 2026–27.
Volume comes from adding thin slices: V=∫ₐᵇ A(x)dx, where A is the cross-sectional area perpendicular to the x-axis. The region in the base determines a length s(x); the stated shape turns that length into area. A square gives A=s². A rectangle of height h gives A=sh; if its height is twice its base, A=2s².
Be explicit about which dimension the base segment supplies. The side length is a distance, so square the entire expression: (top−bottom)², not top²−bottom². With horizontal slices, write A(y) and use dy instead. Area has square units, thickness has length units, and the integral has cubic units. A sketch of a single slice usually resolves more confusion than a complicated three-dimensional drawing.
Let's work through it
- The base is between y=x and y=x² for 0≤x≤1; square slices are perpendicular to the x-axis. Their side is s=x−x². Reason: The base segment’s vertical length supplies the square side.
- Volume is ∫₀¹(x−x²)²dx. Reason: Slice area is the square of the side length.
- Expand and integrate x²−2x³+x⁴ to obtain 1/30 cubic units. Reason: The polynomial antiderivative evaluates the volume.
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
A rectangle has base s and height 3s. What is its area?
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
A rectangle has base s and height 3s. What is its area?
Worked answer and reason
3s².
Multiply the two dimensions.
Retrieval 2
Why integrate A rather than just a side length?
Worked answer and reason
Volume needs area times thickness.
A side-length integral produces square units, not cubic units.
Catch the tempting mistake
- A square between y=f and y=g has area f²−g².
- Its side is f−g, so its area is (f−g)². The difference of squares describes a different geometry.
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 15 · No calculator · analytic
- Unit 8 problem 16 · No calculator · verbal
For the teacher
- Have students draw and label one square and one rectangle slice. Require dimension analysis before computing.
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.