Consider a large uniform rectangle, length L on the long edge and width W on the short edge. This rectangle is rotating so that the axis of rotation is along the long edge of the rectangle and this axis is horizontal. When the rectangle is at equilibrium, the perpendicular line passing through the axis of suspension and the center of mass is vertical and defines the ? = 0 position. This is the position we assume for the calculation below.
Calculate both the horizontal and vertical position of the center of mass by direct integration. This means doing two integrals.
Calculate the moment of inertial of the rectangle, which means doing one integral.