I'd first calculate it the regular way using the formula:
Me, I'd use complex coordinates. The equation for the top curve is . You can do the bottom one. Thus, the parametric form of the top curve is . Now, just substitute both expressions, making sure you get the starting and ending t times correctly, into the formula above.
After I get the right answer, then I'd go on to calculate the moment of the figure and then the y-coordinate of the centroid and see if I get agreement.