A solid of revolution is formed when a plane region is rotated around a line, creating a three-dimensional object whose volume can often be found by integration. The axis of rotation does not have to be the x-axis or y-axis, so the distance from the region to the axis becomes especially important. This idea appears in engineering, physics, and design whenever a shape is produced by spinning a cross section around a shifted center line.
Correct setup depends on choosing slices that match the geometry of the rotation.
Key Facts
- Disk method: V = pi integral from a to b of R(x)^2 dx
- Washer method: V = pi integral from a to b of (R(x)^2 - r(x)^2) dx
- Shell method with vertical shells: V = 2pi integral from a to b of radius(x) height(x) dx
- For rotation about y = k, a vertical slice has radius equal to vertical distance from the curve to y = k.
- For rotation about x = h, a horizontal slice has radius equal to horizontal distance from the curve to x = h.
- Example radius shift: rotating y = f(x) about y = k gives R(x) = |f(x) - k| for a disk edge.
Vocabulary
- Axis of rotation
- The fixed line around which a two-dimensional region is rotated to form a three-dimensional solid.
- Radius function
- A function that gives the distance from a slice of the region to the axis of rotation.
- Disk method
- A volume method used when slices perpendicular to the rotation axis form solid circular disks.
- Washer method
- A volume method used when perpendicular slices form rings with an outer radius and an inner radius.
- Shell method
- A volume method used when slices parallel to the rotation axis form thin cylindrical shells.
Common Mistakes to Avoid
- Using the curve value as the radius when the axis is shifted. The radius must be the distance from the curve to the line of rotation, such as f(x) - k or k - f(x).
- Forgetting the inner radius in a washer setup. A hole in the solid means the volume is outer disk volume minus inner disk volume, not just pi integral R^2.
- Choosing dx or dy without matching the slice direction. Perpendicular slices lead to disks or washers, while parallel slices lead to shells.
- Dropping absolute distance when finding a shell radius. A radius cannot be negative, so use the positive distance between the shell and the rotation line.
Practice Questions
- 1 Find the volume when the region between y = x, y = 0, x = 0, and x = 2 is rotated about the line y = -1 using washers.
- 2 Find the volume when the region under y = 4 - x^2 above the x-axis from x = 0 to x = 2 is rotated about the line y = 5 using washers.
- 3 A region is bounded by y = x^2 and y = 2x. It is rotated about the vertical line x = 3. Explain whether shells using dx or washers using dy would likely give a simpler setup, and justify your choice using radius and height.