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CalculusBy Educator Today

C53 · Unit C8 Applications of integration

Vertical and shifted axes of rotation

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 8.10, 8.12
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Disk and washer slices are perpendicular to the rotation axis. Around a vertical axis, horizontal slices usually require integrating with respect to y. A radius is horizontal distance from that axis to a boundary. For a shifted axis x equals h, compute each distance separately, identify outer and inner radii, and square before subtracting.

Worked example

The region between x equals y squared and x equals 2y for 0 is less than or equal to y is less than or equal to 2, rotated about the y-axis, gives V equals pi ∫ sub 0 squared [(2y) squared minus (y squared) squared ]dy equals 64 pi /15. About x equals minus 1, the radii become 2y+1 and y squared +1. The slice thickness remains dy.

Explore the mathematics

Vertical and shifted axes of rotation

Measure a disk radius from a shifted rotation axis.

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=−1Interval 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 distance from y=−1 to y=x is x+1, so each disk has area π(x+1)².
  3. Add thin slice volumes by integrating their cross-sectional areas.
  4. The volume is 15.3153 cubic units. The shifted axis changes the radius; using x would give the wrong volume.
Vertical and shifted axes of rotation · visual worked method
Vertical and shifted axes of rotation · visual worked methodThe region between x=y² and x=2y for 0≤y≤2, rotated about the y-axis, gives V=π∫₀²[(2y)²−(y²)²]dy=64π/15. About x=−1, the radii become 2y+1 and y²+1. The slice thickness remains dy.Worked example · 1/2The region between x=y² and x=2yfor 0≤y≤2, rotated about they-axis, givesV=π∫₀²[(2y)²−(y²)²]dy=64π/15.About x=−1, theWorked example · 2/2radii become 2y+1 and y²+1. Theslice thickness remains dy.

The region between x=y² and x=2y for 0≤y≤2, rotated about the y-axis, gives V=π∫₀²[(2y)²−(y²)²]dy=64π/15. About x=−1, the radii become 2y+1 and y²+1. The slice thickness remains dy.

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

A common misconception

Keeping dx simply because the original plot is in x-y coordinates can produce slices parallel to the axis, which are not disks or washers.

Investigate and explain

Draw the same region rotated around x equals 0 and x equals minus 1. Label one representative washer in each.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.