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 an axis perpendicular to the axis of rotation. It involves slicing the solid into cylindrical shells and summing their volumes.

5 Must Know Facts For Your Next Test

1. The formula for the volume using cylindrical shells is $$V = \int_{a}^{b} 2\pi x f(x) \, dx$$ for rotation around the y-axis.
2. Cylindrical shells are useful when the function is easier to integrate with respect to $x$ rather than $y$ or vice versa.
3. The height of each cylindrical shell is given by the value of the function at that point, $f(x)$ or $f(y)$.
4. The radius of each cylindrical shell is determined by the distance from the axis of rotation, typically $x$ when rotating around the y-axis, and $y$ when rotating around the x-axis.
5. When setting up an integral using cylindrical shells, ensure that all dimensions (height, radius, and thickness) are correctly expressed in terms of a single variable.

Review Questions

• How do you determine the radius and height of a cylindrical shell?
• Write down the integral expression for finding volumes using cylindrical shells when rotating around the y-axis.
• Why might one choose to use cylindrical shells over disk/washer methods in certain problems?

Related terms

Disk Method: A technique for finding volumes by slicing a solid perpendicular to its axis of revolution, creating disks.

Washer Method: An extension of the disk method used when there is a hollow region within the solid, creating washers instead of disks.

$2\pi$ Factor: $2\pi$ appears in cylindrical shell integrals as it represents the circumference of each shell.