Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
Determining Volume by Slicing/Disk/Washer Method
Objectives
- Determine the volume of a solid by integrating a cross-section (the slicing method).
- Find the volume of a solid of revolution using the disk method.
- Find the volume of a solid of revolution with a cavity using the washer method.
Summary
We can use a definite integral to find the volume of a three-dimensional solid of revolution that results from revolving a two-dimensional region about a particular axis by taking slices perpendicular to the axis of revolution which will then be circular disks or washers.
If we revolve about a vertical line and slice perpendicular to that line, then our slices are horizontal and of thickness . This leads us to integrate with respect to ,as opposed to with respect to ,when we slice a solid vertically.
If we revolve about a line other than the or axis, we need to carefully account for the shift that occurs in the radius of a typical slice. Normally, this shift involves taking a sum or difference of the function along with the constant connected to the equation for the horizontal or vertical line; a well-labeled diagram is usually the best way to decide the new expression for the radius.