$A_1 = \frac{1}{2}(6)(9) = 27 \, \text{ in}^2$

$x_1 = \frac{1}{3}(6) = 2 \, \text{ in}$

$y_1 = \frac{2}{3}(9) = 6 \, \text{ in}$

$A_2 = \frac{1}{2}\pi (3^2) = 14.14 \, \text{ in}^2$

$x_2 = r = 3 \, \text{ in}$

$y_2 = 9 + \dfrac{4(3)}{3\pi} = 10.27 \, \text{ in}$

$A = A_1 + A_2 = 27 + 14.14$

$A = 41.14 \, \text{ in}^2$

$A \bar{x} = \Sigma ax$

$41.14 \, \bar{x} = 27(2) + 14.14(3)$

$\bar{x} = 2.34 \, \text{ in}$

$A \bar{y} = \Sigma ay$

$41.14 \, \bar{y} = 27(6) + 14.14(10.27)$

$\bar{y} = 7.47 \, \text{ in}$

Coordinates of the centroid is at (2.34, 7.47). *answer*