Cross-section

Definition

A cross-section is the shape obtained by cutting a solid object with a plane. It is used to determine the volume of solids by integrating the areas of these cross-sections along an axis.

5 Must Know Facts For Your Next Test

1. Cross-sections are typically taken perpendicular to an axis, such as the x-axis or y-axis.
2. The area of each cross-section can be expressed as a function of the position along the axis (e.g., A(x)).
3. To find the volume of a solid using cross-sections, integrate the area function over the interval defining the solid.
4. Common shapes for cross-sections include circles, rectangles, and triangles.
5. In problems involving volumes by slicing, it's essential to correctly identify and set up the limits of integration.

Review Questions

• How do you determine the area of a cross-section at a given point along an axis?
• What is the integral formula for finding volumes using cross-sectional areas?
• Can you describe how to set up an integral for a solid with circular cross-sections?

Related terms

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

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

Slicing Method: \text{A technique for finding volumes where a solid is divided into thin slices, and their combined volume is found through integration.}