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CalculusBy Educator Today

C36 · Unit C8 Applications of integration

Volumes by cross-sections, disks and washers

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 8.7–8.12
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Volume is ∫A(slice) d(position). With known cross-sections, express the area in terms of the base slice: a square on width w has area w squared, an equilateral triangle has square root of 3w squared /4. Rotation gives disks pi R squared or washers pi (R squared minus r squared). Radii are distances to the rotation axis, which may be shifted away from a coordinate axis.

Worked example

Rotate the region under y equals x on 0 is less than or equal to x is less than or equal to 2 about the x-axis. Each disk has radius x, so V equals pi ∫ sub 0 squared x squared dx equals 8 pi /3. About y equals minus 1, the same region uses outer radius x+1 and inner radius 1; V equals pi ∫ sub 0 squared [(x+1) squared minus 1]dx equals 20 pi /3.

Explore the mathematics

Volumes by cross-sections, disks and washers

Turn a rotating radius into disk area before integrating volume.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict the area of one slice before finding volume.Rotate y=x about y=0Interval 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. The radius is x, so each disk has area πx².
  3. Add thin slice volumes by integrating their cross-sectional areas.
  4. The volume is 3.5343 cubic units. Always measure radius from the rotation axis.
Volumes by cross-sections, disks and washers · visual worked method
Volumes by cross-sections, disks and washers · visual worked methodRotate the region under y=x on 0≤x≤2 about the x-axis. Each disk has radius x, so V=π∫₀²x²dx=8π/3. About y=−1, the same region uses outer radius x+1 and inner radius 1; V=π∫₀²[(x+1)²−1]dx=20π/3.Worked example · 1/2Rotate the region under y=x on0≤x≤2 about the x-axis. Each diskhas radius x, so V=π∫₀²x²dx=8π/3.About y=−1, the sameWorked example · 2/2region uses outer radius x+1 andinner radius 1;V=π∫₀²[(x+1)²−1]dx=20π/3.

Rotate the region under y=x on 0≤x≤2 about the x-axis. Each disk has radius x, so V=π∫₀²x²dx=8π/3. About y=−1, the same region uses outer radius x+1 and inner radius 1; V=π∫₀²[(x+1)²−1]dx=20π/3.

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

A common misconception

Use the difference of squares for washer areas, not the square of the difference. A radius cannot be signed height without considering the axis.

Investigate and explain

Build three slices of one solid using paper. Label the slice width separately from its area and the differential thickness.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.