Let m > 0. Consider the region bounded by y=xm, y=xm+1, x=0, and x=1.
You can (and should) find the center of mass of this region (or, if you
prefer, the centroid of the region) to be the point
with coordinates
The intersection of the moving vertical and horizontal lines in the animation mark the center of mass as m varies over the range from 0.1 to 4.02. You can see that as long as m is small enough, the center of mass is inside the region.
When
the center of mass will be on the upper curve. Solving this equation to find this critical value can be done using numerical methods. It turns out that this critical value is
m=2.369724658859673265580878... .
Back to Dr. Conroy's Math 125 site.