Find the centroid of the first quadrant portion of the disk
| x^{2} + y^{2} <= a^{2} |
Find the centroid of a solid cone having radius a and altitude h for given positive constants a and h.
Hint: Such a solid cone may be realized, up to congruence, as the set of points (x, y, z) in R^{3} for which 0 <= z <= h and
| {r}/{a} + {z}/{h} <= 1 , |
where r^{2} = x^{2} + y^{2}. In this realization the lateral surface of the cone is the surface of revolution obtained by rotating the line
| {x}/{a} + {z}/{h} = 1 |
about the z-axis. A point on that surface is characterized by
| {r}/{a} + {z}/{h} = 1 . |