Theorem of Pappus for volume

Definition

Theorem of Pappus for volume states that the volume of a solid of revolution generated by rotating a plane region about an external axis is equal to the product of the area of the region and the distance traveled by its centroid.

5 Must Know Facts For Your Next Test

1. The theorem applies to solids generated by rotating a plane figure around an axis external to the figure.
2. The volume $V$ is given by $V = A \cdot D$, where $A$ is the area of the region and $D$ is the distance traveled by its centroid.
3. To find $D$, you need to know the centroid's coordinates and understand how it moves during rotation.
4. $D$ can be computed as $2\pi r$, where $r$ is the distance from the centroid to the axis of rotation.
5. The theorem simplifies complex volume calculations into more manageable geometric problems involving centroids.

Review Questions

• What is the formula for computing volume using Theorem of Pappus for volume?
• How do you determine the distance traveled by a centroid in Theorem of Pappus?
• Explain why knowing the location of a centroid is crucial when applying Theorem of Pappus.

Related terms

Centroid: The point representing the center of mass or geometric center of a plane figure.

Solid of Revolution: A solid formed by rotating a plane region around an external axis.

Area: $A$ represents the size of a two-dimensional surface within boundaries.