Theorems of Pappus and Guldinus:

There are two theorems created by Pappus and Guldinus. First theorem is useful in calculating the surface of revolution of a given curve rotating around a given axis which does not intersect with the curve. The second theorem associated to the determination of the solid-volume of revolution of a given area rotating around a given non-intersecting axis.

Theorems

Theorem I

The area of a surface produced by revolving a plane curve around a non-intersecting axis in the plane of the curve is equal to the product of (a) the length of the curve, and (b) the distance travelled by the centroid G of the curve throughout the revolution.

Theorem II

The volume of the solid generated by revolving a plane area APBA in Figure around a non-intersecting axis in its plane is equivalent to the product of (a) the area APBA, and (b) the length of the path travelled by the centroid G of the area throughout the rotation around the axis.