To locate the center of mass, you'll need the area of the region, the moment about the x-axis and the moment about the y-axis. Use the hint your problem gave to solve for the area. The formula for the moments should be in your book, if not look here
Consider the thin plate, of constant density p , which lies above the x-axis and between the two ellipses Find its center of mass(HINT: In order to reduce the amount of integration required, use symmetry and the formula which states that the area A of the region enclosed by the ellipse )