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Chapter 6 | Applications of Integration
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 π units 2 . 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 π units. Then, the volume of the torus is A · d =32 π 2 units 3 .
6.34 Let R be a circle of radius 1 centered at (3, 0). Use the theorem of Pappus for volume to find the volume of the torus generated by revolving R around the y -axis.
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