> > One needs to find the centroid, which is the center > of mass or the average x coordinate and the average > y- coordinate of the region, (x bar, y bar) in 2-space. > Calculus may come into play here, but the centroid > usually is the center when that label is used, like for > ellipses, rectangles, squares, equilateral triangles, > etc. Then one proceeds as follows. > > Find the distance D between centroid and axis (line) > of revolution, using a formula for the distance between > a point and a line. Pappus' formula for volume is > V = 2pi*D*A where A is the area of the region. >
One can also use Pappus' theorem in the reverse manner, to *find* the centroid of certain regions. For example, suppose you want to find the centroid of a semi-circular region (where one side of the region is the diameter AOB). By symmetry, the centroid must lie somewhere on OC, the perpendicular bisector of AB. When the semi-circular region is rotated to form a solid of revolution, the result is a solid sphere with volume given by v = (4/3)*PI*r^3., and the area of the region is a = (1/2)*PI*r^2. Plug these values into Pappus' formula v = 2*PI*d*a and solve for d, which locates the position of the centroid on OC.