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 around an external axis is equal to the product of the area of the region and the distance traveled by its centroid. It applies to both horizontal and vertical rotations.

5 Must Know Facts For Your Next Test

1. The formula for using the theorem is $V = A \cdot d$, where $A$ is the area of the plane region and $d$ is the distance traveled by its centroid.
2. To find $d$, you need to calculate $2\pi R$, where $R$ is the distance from the centroid to the axis of rotation.
3. It is crucial that the axis of rotation does not intersect with the shape being rotated.
4. Pappus' theorem can be used for complex shapes, simplifying volume calculations when integrating would be challenging.
5. Both horizontal and vertical rotations are accommodated, but each requires identifying different centroids.

Review Questions

• What formula does Pappus' Theorem for Volume use?
• How do you determine the distance traveled by a shape's centroid in Pappus' Theorem?
• Can Pappus' Theorem be applied if the axis of rotation intersects with the shape?

Related terms

Centroid: The point that represents the geometric center or average position of all points in a shape or object.

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

$2\pi R$: $\text{Represents} \text{the circumference path taken by a point at radius} \ R \ \text{when it rotates around an axis.}$