Area and Volume of Revolution


The surface area and volume of any body of revolution can be found according to the theorems of Pappus and Guldinus, which are stated as follows:

If a curve is revolved about a nonintersecting axis, the area of the resulting surface of revolution is equal to the length of the revolving curve times the distance traveled by the centroid of the curve.

equation (1)

where Ais the surface of revolution, r-bar is the perpendicular distance from the centroid of the curve to the axis of revolution θ is the angle of revolution measured in radians, and Lis the length of the revolving curve.

 figure

If an area is revolved about a nonintersecting axis, the volume of the resulting surface of revolution is equal to the area of the revolving surface times the distance traveled by the centroid of the area.

equation (2)

where V is the volume of revolution, and A is the revolving area.

The application of the theorems of Pappus and Guldinus is demonstrated in the following examples.

Example 1

Example 2

 figure