Calculate the moments M_y and M_x and the center of mass of a lamina with density p=1 bounded by the curves y = 5 -|x| and y = 1 + |x|.

Answer 1

Since the slopes of the four line segments are all either +1 or -1, it is easy to locate the center of mass at the point (0,3), and determine that the shape is a square centered at that point.

The moments M_x and M_y through the CM will be equal. by the symmetry of a square.

M_x = 2 ∫ x² [4 - 2x] dx with limits of integration 0 and 2. {That is twice the contribution from the right side of the square}

The integral is easy to evaluate giving M_x = 32/3