2. (7+4+2 points) The center of mass or centroid of a shape is the unique point C such that if you let the shape hang freely from any point on its outside, 0' will always lie directly below it. The :z-coordinate of the centroid of a shape 3' of uniform density is the average z-coordinate of all points in S, and analogously for the y-ooordinate and {when applicable) z-ooordinate of S. For each of the following shapes (with uniform density), determine its centroid: (a) The portion of the unit disk {the interior and boundary of the unit circle) lying above y=-%- (b) The cube with eight vertices {:I, :|:1, i1), minus the portion of it with .13 S 0, y E 0, and z E 0 simultaneously (c) The intersectlon of a sphere of radius 5 centered at the orlgin With a cube of SldB length 2 centered at the origin. Hint: You can use symmetry to your advantage in many cases. You are also welcome to use elementary geometry where helpful