Theorem of Pappus Problem: Pappus was a Greek mathematician who lived in Alexandriain the fourth century A.D. One of his theoremsis stated here.Theorem: The Theorem of Pappusfor Volumeswhere A is the area of the region beingrotated, is the displacement from the axisof rotation to the centroid of the region, andthe region is not on both sides of the axis ofrotation. The quantity 2 is thus thedistance the centroid travels as the regionrotates.The volume, V, of a solid of revolution isgiven byIn 14, you saw an example of thistheorem. In this problem you will use thetheorem, once forward and once backward.a. Toroid Problem: A toroid (Figure 11-4q) isformed by rotating a circle of radius r aboutan axis R units from the center of the circle,where r R. Find the volume of the toroid.Figure 11-4qb. Centroid of a Semicircle: A semicircle ofradius r is rotated about its diameter toform a sphere (Figure 11-4r). You knowformulas for the area of a semicircle and forthe volume of a sphere. Use these facts tofind the displacement from the center of asemicircle to its centroid.Figure 11-4r

Study Guide for Midterm Exam II I. Major Ideas of Readings 1. Coflan: discusses balance of trade (deficit and surplus), China’s trading partners have seen decreases in surpluses or growing deficits 2. Cowen: “Lack of Major Wars May Be Hurting Economic Growth” but helps other things (environment, peaceful relations, social tolerance, etc.) 3. Slaughter: comparative advantage andAmerican...