Disk method

Definition

The disk method is a technique used to determine the volume of a solid of revolution by integrating the cross-sectional area of disks perpendicular to an axis of revolution. The formula involves integrating $\pi [f(x)]^2$ or $\pi [g(y)]^2$ over a given interval.

5 Must Know Facts For Your Next Test

1. The disk method is used when the solid is revolved around the x-axis or y-axis.
2. The volume formula for revolving around the x-axis is $V = \pi \int_a^b [f(x)]^2 \, dx$.
3. For rotation around the y-axis, the formula becomes $V = \pi \int_c^d [g(y)]^2 \, dy$.
4. The function being integrated represents the radius of the disks formed by slicing perpendicular to the axis of revolution.
5. Ensure that functions are continuous and properly bounded within the interval for accurate results.

Review Questions

• What is the integral form for calculating volume using the disk method around the x-axis?
• How do you set up an integral for finding volume with respect to y using disks?
• Explain why it’s essential for functions involved in disk method calculations to be continuous over their intervals.

Related terms

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

Washer Method: Similar to the disk method but used when there is a hole in the middle, requiring subtraction of volumes.

Cylindrical Shells Method: $A$ method for finding volumes by integrating cylindrical shells, useful when revolving around axes not aligned with standard coordinates.