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How to use the washer method to find the volume of a shape that is obtained by rotating an appropriate plane region about the x- or y-axis.

The formula for the volume of the solid of revolution that has washers as its cross section is given by
  • π∫abrox2-rix2dx, if the axis of rotation is the x-axis.
  • π∫abroy2-riy2dy, if the axis of rotation is the y-axis.
Note that ro gives the radius of the outer region of the washer and ri gives the radius of the inner region. Also note that ro (x) and ri (x) are the radii of the washers rotated about the x-axis and that ro (y) and ri (y) are the radii of the washers rotated about the y-axis.

*The washer method is an application of the method of slicing.

Artists, graphic designers, biologists, and other professionals often use the washer method to calculate the volume of hollow objects. Here are some examples.

Glassblowers create custom glass art pieces by inflating molten glass into a bubble and rotating it at the end of a blowpipe. The glassblower must calculate the volume of glass needed to form a specific shape. This must be done ahead of time to know how much glass to use.
To estimate the total biomass of jellyfish in the ocean, a biologist must know the volume of each individual jellyfish. Once the volume is estimated, the mass can be calculated by multiplying this volume by the density of a jellyfish.
Graphic designers must calculate the volumes of solid and hollow objects to construct 3D animated worlds. The volumes of the hollow domes can be calculated using the washer method, and the volumes of the solid pillars can be calculated using the disk method.

Click on the tabs on the right to watch videos on rotating a region about the x-axis and y-axis.

VOLUMES OF REVOLUTION - WASHER METHOD
To rotate grid, click and drag
REVOLVE REGION
# OF SLICES
AXIS OF REVOLUTION
x y
Region 1     y1=0.5,x-0.5,0≤x≤11<x<3y2=1,x,0≤x≤11<x≤3
Region 1     x1=0,y,0.5≤y<1.01.0≤y≤3.0x2=y+0.5,3,0.5≤y≤2.52.5<y≤3.0
Volume  ∫abπy22-y12dx≈
Volume  ∫abπx22-x12dx≈
0
Region 2     y=1-(x-1.5)2,      0.5≤x≤2.5
Region 2     x1=1.5-1-y2,  -1≤y≤1      x2=1.5+1-y2,  -1≤y≤1
Volume  ∫abπy2dx≈
Volume  ∫abπx22-x12dx≈
0
Region 3     y1=x,  0≤x≤1y2=x2,0≤x≤1
Region 3     x1=y2, 0≤y≤1x2=y, 0≤y≤1
Volume  ∫abπy22-y12dx≈
Volume  ∫abπx22-x12dx≈
0
Number of Slices:0Width of Slices:0.00Sum of Volume of Slices ≈ 0.00   units3
Click the Revolve Region button and watch the animation before using the slider bars.
1 2 3
1. Revolve the selected region about the x-axis, and then move the second slider to obtain 12 partitioning washers. What is the width of each washer?
2. Revolve the selected region about the y-axis, and then move the second slider to obtain 10 partitioning washers. What is the width of each washer?
3. We want to revolve the selected region about the x-axis to obtain a funnel made of glass. You can consider the funnel as being made up of two pieces. The first piece is the straight stem for
. Set up an integral to compute the volume of the material that makes up the glass stem. Check all that apply.
4. The second piece is the funnel section for . Set up an integral to compute the volume of the material that makes up the glass funnel section. Check all that apply.
5. The total volume of the glass material is equal to the volume of the stem plus the volume of the funnel section. Which of the following describes the total volume of glass material? Check all that apply.
6. Keep the selected region revolved about the x-axis. What is the estimated volume of the object using eight partitioning washers?
7. Now revolve the region about the y-axis. What is the estimated volume of the object using eight partitioning washers?
Click the Revolve Region button and watch the animation before using the slider bars.
1 2 3
1. Revolve the selected region about the x-axis so that a sphere is formed. Move the second slider to obtain eight partitioning washers. What is the width of each washer?
2. Revolve the selected region about the y-axis so that a donut shape is formed. Move the second slider to obtain 20 partitioning washers. What is the width of each washer?
3. We first want to revolve the selected region about the x-axis to obtain a beach ball shape. This problem can be solved using the disk method. Set up an integral to compute the volume of the beach ball. Check all that apply.
4. We now want to revolve the selected region about the y-axis to obtain a donut shape. This problem must be solved using the washer method. Set up an integral to compute the volume of the donut. Check all that apply.
5. Keep the selected region revolved about the y-axis. What is the estimated volume of the object using 10 partitioning washers?
6. Now revolve the region about the x-axis. What is the estimated volume of the object using 10 partitioning washers?
Click the Revolve Region button and watch the animation before using the slider bars.
1 2 3
1. Notice that the lines describing the inner and outer surfaces are symmetric about the line y = x. Revolve the selected region about the y-axis and then the x-axis. Which of the following statements is true?
2. Revolve the selected region about the x-axis and then move the second slider to obtain 20 partitioning washers. What is the width of each disk? Check all that apply.
3. Keep the selected region revolved about the x-axis. What is the estimated volume of the object using 20 partitioning washers?
4. Keep the selected region revolved about the x-axis. What is the estimated volume of the object using 15 partitioning washers?
5. Now revolve the region about the y-axis. What is the estimated volume of the object using 15 partitioning units?
Result