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CalculusBy Educator Today

C52 · Unit C8 Applications of integration

Non-square cross-sections

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 8.7–8.8
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

The base region determines a length, while the cross-section shape determines how that length becomes area. For an equilateral triangle of side w, area is square root of 3w squared /4. For a semicircle whose diameter is w, area is pi w squared /8. For a rectangle, identify the relationship between its height and base explicitly. Then integrate area across the base region.

Worked example

A solid has base between y equals x and y equals x squared on [0,1]. Vertical base length is w equals x minus x squared. Equilateral-triangle sections give V equals ( square root of 3 over 4)∫ sub 0 to the power 1 (x minus x squared) squared dx equals square root of 3 over 120 because the polynomial integral is 1 over 30.

Explore the mathematics

Non-square cross-sections

Use semicircular slice area when the specified length is a diameter.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict the area of one slice before finding volume.Diameter x → radius x/2Interval 0≤x≤1.5

Predict before continuing

Predict the area of one slice before finding volume.

Read the complete explanation
  1. Predict the area of one slice before finding volume.
  2. A semicircle with diameter x has area ½π(x/2)²=πx²/8.
  3. Add thin slice volumes by integrating their cross-sectional areas.
  4. The volume is 0.4418 cubic units. Using πx² would confuse diameter with radius and omit the half.
See the idea: Non-square cross-sections
See the idea: Non-square cross-sectionsThe region lies between the upper line y=x and lower parabola y=x² on [0,1]. A vertical slice has width w=x−x². For equilateral-triangle cross-sections, slice area is √3 w²/4. Integrating gives volume √3/120; the base area is not the volume.xy000.250.250.50.50.750.7511

The region lies between the upper line y=x and lower parabola y=x² on [0,1]. A vertical slice has width w=x−x². For equilateral-triangle cross-sections, slice area is √3 w²/4. Integrating gives volume √3/120; the base area is not the volume.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

For semicircles the base length is a diameter unless the problem says otherwise. Treating it as radius multiplies area by four.

Investigate and explain

Construct a square, semicircle and equilateral triangle on the same segment. Compare their areas before writing the corresponding volumes.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.