Washer method

Definition

The washer method is a technique used to find the volume of a solid of revolution when the region being revolved around an axis creates a hollow shape. It involves integrating the area of washers, which are disks with holes in the center.

5 Must Know Facts For Your Next Test

1. The washer method is used when the solid has a gap or hole along the axis of rotation.
2. The formula for the volume using the washer method is $$V = \pi \int_{a}^{b} [R(x)^2 - r(x)^2] \, dx$$ where $R(x)$ and $r(x)$ are the outer and inner radii, respectively.
3. Ensure that $R(x) \geq r(x)$ over the interval [a, b] to avoid negative volumes.
4. If rotating around the y-axis, express $x$ as a function of $y$ and integrate with respect to $y$.
5. The washer method can be applied to both horizontal and vertical axes of rotation.

Review Questions

• What is the difference between using disks and washers in volume calculations?
• How do you set up an integral for finding volume using the washer method if revolving around the y-axis?
• Why must you ensure that $R(x) \geq r(x)$ when applying the washer method?

Related terms

Disk Method: A technique for finding volumes by slicing perpendicular to an axis of rotation, creating solid disks without holes.

Solid of Revolution: A three-dimensional object obtained by rotating a two-dimensional region around an axis.

Cylindrical Shells Method: A technique for finding volumes by slicing parallel to an axis of rotation, forming cylindrical shells.