Washer method

Definition

The washer method is a technique used to find the volume of a solid of revolution when the solid has a hole in the middle. It involves integrating the difference between the outer radius and inner radius squared, multiplied by $\pi$.

5 Must Know Facts For Your Next Test

1. The washer method is typically used when the solid of revolution is generated by rotating a region around an axis that results in a hollow object.
2. The formula for 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 $R(x)$ and $r(x)$ are correctly identified based on the axis of rotation and positions of functions involved.
4. The limits of integration $[a,b]$ should match the region being revolved around the axis.
5. A common mistake is forgetting to square both radii before subtracting and integrating.

Review Questions

• What is the key difference between the disk method and the washer method?
• How do you determine which function represents $R(x)$ and which represents $r(x)$?
• Write down and explain each part of the formula for calculating volume using the washer method.

Related terms

Disk Method: A technique for finding volumes of solids of revolution when there is no hole, using disks perpendicular to the axis of rotation.

Volume by Slicing: A general approach in calculus to find volumes by summing up infinitesimally thin cross-sectional areas.

Solid of Revolution: \text{A three-dimensional shape generated by rotating a two-dimensional region around an axis.}