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Calculus practice

Volumes Using Cylindrical Shells

Practice exact shell-method volumes from quadratic and cubic regions rotated about vertical or horizontal axes.

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Use cylindrical shells to find each exact volume. Show radius, positive height, setup, and evaluation.

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Build shells from real geometry

Each supplied graph shows the bounded region and its rotation line. Students measure radius to the axis, use a positive lobe height, and integrate every shell contribution exactly.

Learning objective

What this activity builds

Build radius and positive-height functions on every lobe, then evaluate the cylindrical-shell integral exactly.

Worked example

What is the shell-method volume element?

2π(radius)(height) dt.

The shell radius is the distance to the rotation line, while shell height is the positive distance from the curve to the dependent-coordinate zero axis.

Teaching notes

Make the practice count

  • Measure radius from the named rotation line.
  • Make height positive on each lobe.
  • Split a cubic at its middle root before integrating.

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Math SheetNameScoreTeacherDateVolumes Using Cylindrical ShellsUse cylindrical shells to find each exact volume. Show radius, positive height, setup, and evaluation.1. Rotate the shaded region about the vertical line x = 0.y =-12·x+4·x+2·x+12Region: between this curve and y = 0 from x = -4 to -1/2.-4-20-2-11x = 0xyShow shell radius, positive height, setup, and exact volume:2. Rotate the shaded region about the vertical line x = 0.y =2·x-32·x-3Region: between this curve and y = 0 from x = 3/2 to 3.0123-1x = 0xyShow shell radius, positive height, setup, and exact volume:Exact volume; cubic units.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateVolumes Using Cylindrical ShellsUse cylindrical shells to find each exact volume. Show radius, positive height, setup, and evaluation.3. Rotate the shaded region about the vertical line x = 0.y =-1·x+32·x+1·x+0Region: between this curve and y = 0 from x = -3/2 to 0.-2-10-0.50.51x = 0xyShow shell radius, positive height, setup, and exact volume:4. Rotate the shaded region about the vertical line x = 0.y =12·x-32·x-4Region: between this curve and y = 0 from x = 3/2 to 4.024-1.5-1-0.50.5x = 0xyShow shell radius, positive height, setup, and exact volume:Exact volume; cubic units.math-sheet.comPage 2 of 2

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