Solid of revolution

Definition

A solid of revolution is a three-dimensional object obtained by rotating a two-dimensional region around an axis. The volume of such solids can be calculated using integration techniques.

5 Must Know Facts For Your Next Test

1. The Disk Method is used when the solid is formed by rotating a region around the x-axis or y-axis, creating cylindrical disks.
2. The Washer Method extends the Disk Method to accommodate holes in the middle of the solid, resulting in washers instead of disks.
3. The Shell Method involves slicing the solid into cylindrical shells and is particularly useful when rotating around an axis parallel to the boundaries of the region.
4. A common formula for volume using the Disk Method is $V = \pi \int_a^b [R(x)]^2 dx$, where $R(x)$ is the radius of each disk.
5. The Shell Method formula for volume is $V = 2\pi \int_a^b rh dx$, where $r$ is the radius and $h$ is the height of each shell.

Review Questions

• What are the differences between using the Disk Method and the Shell Method?
• How would you set up an integral to find the volume of a solid formed by rotating a function around an arbitrary axis?
• When should you use washers instead of disks in calculating volumes?

Related terms

Disk Method:

A technique for finding volumes by slicing perpendicular to an axis and summing up circular disks.

Washer Method:

An extension of the Disk Method that accounts for solids with holes, resulting in washer-shaped slices.

Shell Method: A technique for finding volumes by slicing parallel to an axis and summing up cylindrical shells.