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Middle of Mass Measured from Moment

The center of mass of an object is the point that lies in the average position of mass in that object. For a normal polygon, that point is in the middle of the shape. However, if the shape is irregular, has concave angles, or has holes, the center of mass can be far from the middle of the object. Two things come into factor when determining the center of mass of an object. Weight and distance. Obviously, if one side of an object has more mass than the other, the center of mass will be on the heavier side. But, distance plays a role, too. If two sides of an object have the same mass, but one is farther from the middle, the center of mass will be on the side that has farther mass. Multiplying both of these factors together gives the moment of an object. Dividing the moment by the overall mass gives the location of the center of mass.
  The reason I chose my shape is to mimic how those toy balancing birds work. The toy is a model bird with its wings outstretched. The bird's beak is pointing downwards and the bird is able to balance on the beak because the center of mass is right on the beak. It does not look like the bird should be able to balance, but because the wings are far from the rest of the body and go in front of the beak a bit, the wings give a big enough moment to make the center of mass right on the beak. My model similarly has outstretched parts that cause the center of mass to be near the point of the triangle in the middle. After making the shape, I realized it looked like the letter M. That led to this post's alliterative title.
  The way to find the center of mass in a two dimensional object is to find the moment in the x direction and in the y direction, then divide by area and you have your coordinates for the center of mass. My shape has many cut corners in the literal sense, so many integrals are needed for my shape. My shape goes from -4 to 4 in both directions. In the y direction, the way to find the moment is to calculate \[\int_{-4}^{-3.5}(7.5+x-(-7.5-x))xdx + \int_{-3.5}^{-2.5}(4+4)xdx + \int_{-2.5}^{-2}(4-(-1.5+x))xdx + \int_{-2}^{-1.5}(4-(3.5+x))xdx +\]\[\int_{-1.5}^{-1}(4-2)xdx + \int_{-1}^{0}(4-(-2x))xdx + \int_{0}^{1}(4-2x)xdx + \int_{1}^{1.5}(4-2)xdx + \int_{1.5}^{2}(4-(3.5-x))xdx + \]\[\int_{2}^{2.5}(4-(-1.5-x))xdx+\int_{2.5}^{3.5}(4+4)xdx + \int_{3.5}^{4}(7.5-x-(-7.5+x))xdx\] This comes out to equal 0. For the x direction, I rotated the shape in order to integrate over x. The moment in the x direction is \[\int_{-4}^{-3.5}(7.5+x-(-7.5-x))xdx + \int_{-3.5}^{-2}(4+4)xdx + \int_{-2}^{-1.5}(4-(3.5+x)+-x/2-x/2+(-3.5-x)+4)xdx + \]\[\int_{-1.5}^{0}(4-2+-x/2-x/2+-2+4)xdx + \int_{0}^{3.5}(4-2+-2+4)xdx + \int_{3.5}^{4}(7.5-x-(-1.5+x)+1.5-x-(-7.5+x))xdx\] The moment in the x direction comes out to be -28.083. The way to find out the area is to take either of the ways to find the moment and take away the extra x in each integral. So, the area is equal to \[\int_{-4}^{-3.5}(7.5+x-(-7.5-x))dx + \int_{-3.5}^{-2}(4+4)dx + \int_{-2}^{-1.5}(4-(3.5+x)+-x/2-x/2+(-3.5-x)+4)dx + \]\[\int_{-1.5}^{0}(4-2+-x/2-x/2+-2+4)dx + \int_{0}^{3.5}(4-2+-2+4)dx + \int_{3.5}^{4}(7.5-x-(-1.5+x)+1.5-x-(-7.5+x))dx\] which comes out to equal 41.5. Dividing the moment in the y direction by the area gives an x coordinate of 0. Dividing the moment in the x direction by the area gives an x coordinate of -.677, but because I rotated the shape to integrate over x, the y coordinate of the center of mass is .677. Overall, the center of mass of my object lies on (0, .677).
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