Volumes of Revolution: Disks, Washers and Shells
Set up a volume as an integral of cross-sections, and choose between slicing perpendicular to the axis and using cylindrical shells based on which integral is easier.
What a learner can do afterwards
- Write the integral for a solid generated by rotating a region about the x-axis
- Use washers when the solid has a hole and identify both radii
- Say when shells give a simpler integral than disks for the same solid
1 · Read
Picture a region under a curve, cut into thin vertical strips. Spin one strip around the x-axis and you get a thin disk. Its radius equals the function value, so its volume is pi times radius squared times dx. For y equals the square root of x from 0 to 4, each disk has radius square root of x, and the square is x, so the volume is pi times the integral from 0 to 4 of x dx. A flat check: under y equals 3 from 0 to 2, every disk has radius 3, which is a cylinder of radius 3.
Maya rotates the region between y equals x on top and y equals x squared below, from 0 to 1, about the x-axis. The gap between the curves leaves a hole, so each slice is a washer, not a full disk. The outer radius follows the top curve, x, and the inner radius follows the bottom curve, x squared. Her volume is pi times the integral from 0 to 1 of x squared minus x to the fourth dx. With numbers alone, outer radius 5 and inner radius 3 give 25 minus 9, which is 16, so the washer area is 16 pi.
Shells wrap the region in nested tubes instead of stacking slices. One shell at radius r with height h gives 2 pi times r times h times dx. Take y equals x squared from 0 to 1 rotated about the y-axis: a vertical strip at x has height x squared, so each shell gives 2 pi times x times x squared, which is 2 pi times x cubed dx. Shells win here because disks would force you to rewrite the curve as x equals the square root of y.
Name your method before you integrate, then fill radius and height from the picture. Run strips across the axis for disks and washers, and along the axis for shells. Square each radius first and then subtract, since squaring the difference gives the wrong area.
Disks fill the solid, washers subtract the hole, and shells wrap it in tubes, so match the method to the picture.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.