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Calculus practice

Volumes of Washers and Disks

Practice exact disk and washer volumes for regions between intersecting polynomial curves, including shifted axes.

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Use washers or disks to find each exact volume. Show the radii, setup, and evaluation.

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Measure radii from the axis

Each graph shows an enclosed region and its rotation line. Students identify positive outer and inner radii before expanding the washer or disk area.

Learning objective

What this activity builds

Identify positive outer and inner radii, expand their squares, and evaluate the washer or disk integral exactly.

Worked example

What is the washer-method volume element?

π(R²-r²) dt.

The outer and inner radii are positive distances from the named rotation line to the two boundary curves.

Teaching notes

Make the practice count

  • Measure both radii from the rotation line.
  • Keep R larger than r regardless of curve order.
  • Use dx for functions of x and dy for functions of y.

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Every worksheet contains four tasks. Choose disk and washer types plus functions of x, y, or both.

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Math SheetNameScoreTeacherDateVolumes of Washers and DisksUse washers or disks to find each exact volume. Show the radii, setup, and evaluation.1. Rotate the shaded region about the horizontal line y = 0.y = F(x) =12·x2-34·x-12y = G(x) =0-1012-1.5-1-0.50.5y = 0xyShow R, r, the washer/disk integral, and exact volume:2. Rotate the shaded region about the horizontal line y = 0.y = F(x) =-12·x2+14·x+2y = G(x) =12-2-101212y = 0xyShow R, r, the washer/disk integral, and exact volume:Exact volume; cubic units.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateVolumes of Washers and DisksUse washers or disks to find each exact volume. Show the radii, setup, and evaluation.3. Rotate the shaded region about the horizontal line y = 7/2.y = F(x) =12·x2+54·x-10y = G(x) =5·x-170241234y = 7/2xyShow R, r, the washer/disk integral, and exact volume:4. Rotate the shaded region about the horizontal line y = 0.y = F(x) =-12·x2+32·x-1y = G(x) =0012-0.50.5y = 0xyShow R, r, the washer/disk integral, and exact volume:Exact volume; cubic units.math-sheet.comPage 2 of 2

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