Solid of revolution

Definition

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional region around an axis. This technique is commonly used to find volumes of solids using integral calculus.

5 Must Know Facts For Your Next Test

1. The volume of a solid of revolution can be calculated using the disk method or the washer method, depending on whether the region has a hole.
2. In the disk method, the volume is found by integrating the area of circular disks perpendicular to the axis of rotation.
3. In the washer method, the volume is found by integrating the area of washers (disks with holes) perpendicular to the axis of rotation.
4. The choice of axis (horizontal or vertical) affects how you set up your integral and which function represents your radius.
5. $V = \pi \int_a^b [R(x)^2 - r(x)^2] dx$ is a common formula for finding volumes using washers, where $R(x)$ and $r(x)$ are outer and inner radii.

Review Questions

• What methods can be used to find the volume of a solid of revolution?
• How does changing the axis of rotation affect your integral setup?
• What is the formula for calculating volume using washers?

Related terms

Disk Method: A technique for finding volumes by slicing a solid into thin disks perpendicular to an axis and integrating their areas.

Washer Method: A technique for finding volumes by slicing a solid with holes into thin washers and integrating their areas.

Axis of Rotation: The line around which a two-dimensional region is rotated to create a solid of revolution.