## Exercises

1. In the case that the density is uniform, show that are independent of and are given by

where V is the volume of the domain D. In this case, the point is called the centroid of the domain D.

2. In class, we treated the case of a plate of uniform thickness whose density could only depend on x and y. Show how the general equations in this lab reduce to the ones we found in class for this special case.
3. Find the centroids of the following geometrical shapes in two dimensions.
1. A half-disk of radius R.
2. An equilateral triangle of side L.
3. A parabolic spandrel. That is, the region D is bounded below by the x axis, above by the curve , and on the right by the vertical line x=a, where k and a are positive constants. Note that your answer will depend on the parameters k and a.
4. Find the centroids of the following geometrical solids in three dimensions.
1. The portion of the unit sphere in the first octant. That is, the portion of the unit sphere for which x, y, and z are non-negative.
2. The region bounded by the paraboloid and the x-y plane.
3. The tetrahedron bounded by the coordinate planes and the plane x+y+z = 1.
5. Find the center of mass of D, if D is the region bounded by , y= 3x, the x-y plane, and if is given by

William W. Farr
Sun Sep 17 18:36:55 EDT 1995