ES190 Theorems of Pappus 
The theorems of Pappus may be used to find the surface of revolution and volume of revolution for bodies generated by revolving a plane curve about an axis. 
THEOREM I: The area of a surface of revolution is equal to the length of the generating curve times the distance traveled by the centroid while the surface is being generated. 
This is shown in the diagram... 
We can write this in equation form as where L is the length of the curve. 
THEOREM II: The volume of a body of revolution is equal to the generating area times the distance traveled by the centroid of the are while the body is generated. 

This is shown in the diagram below: 
We can write this in equation form as where A  generating area. 