Answer
The $y$-coordinate of the centroid: $\bar y = \frac{4}{{3\pi }}a$.
Work Step by Step
We have the region ${\cal D}$ defined by the upper half of the disk ${x^2} + {y^2} \le {a^2}$, $y \ge 0$. So, the area of ${\cal D}$ is $A = \frac{1}{2}\pi {a^2}$.
Since the disk of radius $a$ revolved about the $x$-axis is a sphere of radius $a$, we obtain the volume $V = \frac{4}{3}\pi {a^3}$.
Recall the Pappus's Theorem in Exercise 60:
$V = 2\pi A\bar y$
Substituting $A$ and $V$ in the equation above gives
$\frac{4}{3}\pi {a^3} = 2\pi \left( {\frac{1}{2}\pi {a^2}} \right)\bar y$
$\bar y = \frac{4}{{3\pi }}a$
Thus, the $y$-coordinate of the centroid: $\bar y = \frac{4}{{3\pi }}a$.
