Method of cylindrical shells.

Definition

The method of cylindrical shells is a technique for finding the volume of a solid of revolution by integrating along the axis perpendicular to the axis of rotation. The volume is computed by summing up cylindrical shell volumes formed by revolving a region around an axis.

5 Must Know Facts For Your Next Test

1. The formula for the volume using cylindrical shells is $V = 2\pi \int_{a}^{b} x f(x) \, dx$ when revolving around the y-axis.
2. When revolving around the x-axis, adjust the formula to $V = 2\pi \int_{a}^{b} y f(y) \, dy$.
3. Cylindrical shells are particularly useful when the function is easier to express in terms of $x$ or $y$, depending on the axis of rotation.
4. The height of each cylindrical shell is given by the function value at a particular point.
5. The radius of each cylindrical shell is determined by the distance from that point to the axis of rotation.

Review Questions

• What is the general formula for finding volume using cylindrical shells when rotating around the y-axis?
• How do you determine the height and radius of a cylindrical shell?
• In what scenarios would using cylindrical shells be more advantageous than other methods?

Related terms

Disk Method: A technique for finding volumes of revolution where cross-sections perpendicular to an axis are disks.

Washer Method: An extension of the disk method used when there is a hole in the middle, creating washers as cross-sections.

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