Cross-section

Definition

A cross-section is the intersection of a solid object with a plane, resulting in a two-dimensional shape. It is used to determine volumes by slicing the object into thin slices.

5 Must Know Facts For Your Next Test

1. Cross-sections can be used to approximate the volume of irregular solids by summing the volumes of thin slices.
2. The shape of the cross-section depends on the orientation and position of the slicing plane.
3. Using integration, you can find the volume of a solid by integrating the area of its cross-sections along an axis.
4. $V = \int_{a}^{b} A(x) \, dx$ is the formula for finding the volume using cross-sectional areas $A(x)$ from $x=a$ to $x=b$.
5. Common cross-sectional shapes include squares, rectangles, circles, and triangles.

Review Questions

• What is a cross-section and how does it relate to finding volumes?
• How do you set up an integral to find the volume using cross-sections?
• What are some common shapes that can appear as cross-sections when slicing different solids?

Related terms

Volume: The amount of space occupied by a three-dimensional object, often calculated using integrals in calculus.

Integral: A mathematical concept that represents the accumulation of quantities, such as areas under curves or volumes.

Slicing: A method for determining volume by dividing a solid into thin slices and summing their volumes.