Pappus's theorem for volume gives a fast way to find the volume of many solids of revolution. Instead of setting up a disk, washer, or shell integral, you use the area of the rotating plane region and the distance traveled by its centroid. This matters because it connects geometry, calculus, and physical intuition in one compact formula.
The theorem is especially useful when the centroid of a shape is already known or easy to compute.
Key Facts
- Pappus's volume theorem: V = A(2πR), where A is the area of the plane region and R is the distance from its centroid to the axis.
- The centroid must travel in a circle, so its path length is 2πR.
- The axis of rotation must be external to the region and must not cut through the region.
- If the centroid has coordinates (xbar, ybar), then R is the perpendicular distance from (xbar, ybar) to the axis of rotation.
- Centroid of a rectangle with width w and height h is at its geometric center, so A = wh and R is measured from the center to the axis.
- For a region made from simpler parts, use xbar = Σ(Ai xi)/ΣAi and ybar = Σ(Ai yi)/ΣAi before applying V = 2πRA.
Vocabulary
- Pappus's centroid theorem
- A theorem that finds the volume of a solid of revolution by multiplying a plane region's area by the distance traveled by its centroid.
- Centroid
- The balance point or geometric average position of a plane region.
- Solid of revolution
- A three-dimensional solid formed by rotating a plane region around an axis.
- Axis of rotation
- The fixed line around which a plane region rotates to generate a solid.
- External axis
- An axis of rotation that does not pass through or intersect the rotating region.
Common Mistakes to Avoid
- Using the distance from the edge instead of the centroid is wrong because Pappus's theorem uses the path of the centroid, not the path of a boundary point.
- Applying the theorem when the axis cuts through the region is wrong because the standard volume theorem requires the axis to be external to the region.
- Forgetting the factor 2π is wrong because the centroid travels a full circular distance of 2πR, not just a distance R.
- Using diameter instead of radius for R is wrong because R is the perpendicular distance from the centroid to the rotation axis.
Practice Questions
- 1 A rectangle has width 4 cm and height 3 cm. Its nearest side is 5 cm from a vertical external axis, and the rectangle rotates around that axis. Find the volume of the solid using Pappus's theorem.
- 2 A semicircular region has radius 6 cm and area 18π cm^2. Its centroid is 4r/(3π) from the flat side. If the flat side lies 10 cm from a parallel external axis on the opposite side of the centroid, find the volume generated by rotation.
- 3 Explain why Pappus's theorem cannot be directly applied to a region if the axis of rotation passes through the region.