Theorem of Pappus

Proof

We can prove the case when the region is bounded above by the graph of a function \(f \left(\right. x \left.\right)\) and below by the graph of a function \(g \left(\right. x \left.\right)\) over an interval \(\left[\right. a , b \left]\right. ,\) and for which the axis of revolution is the y-axis. In this case, the area of the region is \(A = \int_{a}^{b} \left[\right. f \left(\right. x \left.\right) - g \left(\right. x \left.\right) \left]\right. d x .\) Since the axis of rotation is the y-axis, the distance traveled by the centroid of the region depends only on the x-coordinate of the centroid, \(\overset{–}{x} ,\) which is

where

Then,

and thus

However, using the method of cylindrical shells, we have

So,

and the proof is complete.

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Example 6.34

Using the Theorem of Pappus for Volume

Let R be a circle of radius 2 centered at \(\left(\right. 4 , 0 \left.\right) .\) Use the theorem of Pappus for volume to find the volume of the torus generated by revolving R around the y-axis.

Solution

The region and torus are depicted in the following figure.

Figure 6.74 Determining the volume of a torus by using the theorem of Pappus. (a) A circular region R in the plane; (b) the torus generated by revolving R about the y-axis.

The region R is a circle of radius 2, so the area of R is \(A = 4 \pi\) units². By the symmetry principle, the centroid of R is the center of the circle. The centroid travels around the y-axis in a circular path of radius 4, so the centroid travels \(d = 8 \pi\) units. Then, the volume of the torus is \(A \cdot d = 32 \pi^{2}\) units³.