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
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.
Find the centroids of the following geometrical solids in three
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
The tetrahedron bounded by the coordinate planes and the plane
x+y+z = 1.
Find the center of mass of D, if D is the region
bounded by , y= 3x, the x-y plane, and
is given by