- 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**. - 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. - Find the centroids of the following geometrical shapes in two dimensions.
- A half-disk of radius
**R**. - An equilateral triangle of side
**L**. - 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**.

- A half-disk of radius
- Find the centroids of the following geometrical solids in three
dimensions.
- 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. - The region bounded by the paraboloid and the
**x-y**plane. - The tetrahedron bounded by the coordinate planes and the plane
**x+y+z = 1**.

- The portion of the unit sphere in the first octant. That is, the
portion of the unit sphere for which
- 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 -

Sun Sep 17 18:36:55 EDT 1995