Formulate the Pappus Theorem Use the theorem to find the vol

Solution

If you know the centroid of a plane figure, you can use Pappus\'s Theorem to find the volume of a solid of rotation of that plane figure. (Conversely, if you know the volume of a solid of rotation, you can reverse-engineer the centroid using Pappus\'s Theorem.) The x coordinate of the centroid is denoted X, and the y coordinate is denoted y. The centroid X of a finite number of point masses is the sum of the product of each mass and its x-coordinate divided by the sum of all the masses. The centroid of a plane figure is the integral of the x-values of all the slices of the area divided by the total area. For example, the area of a quarter circle of radius r is (1/4) p r², so the centroid of a quarter circle given by y=sqrt(r²-x²) is 4/(p r²) ó ô õ r 0 x sqrt(r²-x²) dx = 4/(p r²)(1/3) p r³ = (4/3)(r/p) The centroid of a semicircle is the same, and so this is the value used in the table, below, to calculate the volume of a sphere.