Slicing method

Definition

The slicing method is a technique used to determine the volume of a solid by integrating the cross-sectional area perpendicular to an axis. It involves summing up infinitesimally thin slices of the solid.

5 Must Know Facts For Your Next Test

1. The integral setup for the slicing method generally takes the form $V = \int_a^b A(x) \, dx$, where $A(x)$ is the cross-sectional area at position $x$.
2. The cross-sectional area function $A(x)$ must be determined based on the geometry of the solid.
3. Slices are perpendicular to the axis along which you integrate, typically either the x-axis or y-axis.
4. In cylindrical coordinates, this method can be adapted to integrate with respect to $r$ and $\theta$ for solids of revolution.
5. Common shapes for slicing include disks, washers, and rectangular strips.

Review Questions

• What is the general form of the integral used in the slicing method?
• How do you determine the cross-sectional area function $A(x)$ for a given solid?
• When using the slicing method, along which axis are slices typically taken?

Related terms

Disks: Slices that are circular in shape; used when finding volumes by rotating a region around an axis.

Washers: Slices with a hole in them (ring-shaped); used when finding volumes by rotating around an axis but there is a gap between the region and axis.

Cross-Sectional Area: The area of a slice perpendicular to an axis, denoted as $A(x)$ or $A(y)$ depending on integration variable.