Centroid of a Cylindrical Wedge

Taking a slice of the wedge at $x = t$ , where $-3 \le t \le 3$ of thickness $dt$ , then this slice will be a rectangle of dimensions $dt \times 2 \sqrt{9 - t^2} \times \dfrac{5}{3} (t + 3)$ , and the centroid of the slice is at $( t, 0, \dfrac{5}{6} (t + 3) )$

The centroid $G = \displaystyle \left( \dfrac{1}{V} \right) \int_{t = -3 } ^{ 3 } ( t, 0, \dfrac{5}{6} (t + 3 ) ) A(t) dt$

where $A(t)$ is the area of the slice and is equal to $\dfrac{10}{3} (t + 3) \sqrt{9 - t^2}$ . The volume $V$ is half the volume of the cylinder, $V = 45 \pi$

Using the substitution $t = -3 \cos \theta$ , then $dt =3 \sin \theta d \theta$ , and the integral becomes:

$G = \displaystyle \left( \dfrac{1}{45 \pi} \right) \int_{\theta = 0}^{\pi} ( -3 \cos \theta, 0, \dfrac{5}{2} (1 - \cos \theta) ) \cdot (90) (1 - \cos \theta) \sin^2 \theta d \theta$

Simplifying and expanding,

$G = \displaystyle \left( \dfrac{2}{ \pi} \right) \int_{\theta = 0}^{\pi} ( -3 \cos \theta (1 - \cos \theta) \sin^2 \theta, 0, \dfrac{5}{2} \sin^2 \theta (1 - \cos \theta)^2 ) d \theta$

Now,

$\displaystyle \int_{\theta = 0}^{\pi} - 3 \cos \theta (1 - \cos \theta) \sin^2 \theta d \theta = \int_{\theta = 0}^{\pi} 3 \cos^2 \theta \sin^2 \theta d \theta = \left( \dfrac{3}{4} \right) \int_{\theta = 0}^{\pi} \sin^2 (2 \theta) d \theta = \left( \dfrac{3}{4} \right) \displaystyle \int_{\theta = 0}^{\pi} \dfrac{1}{2}(1- \cos (4 \theta)) d \theta = \dfrac{3 \pi }{8}$

And,

$\displaystyle \int_{\theta=0}^{\pi} \dfrac{5}{2} \sin^2 \theta (1 - \cos \theta)^2 d \theta = \dfrac{5}{2} \int_{\theta=0}^{\pi} \sin^2 \theta - 2 \cos \theta \sin^2 \theta + \cos^2 \theta \sin^2 \theta d \theta$

$= \dfrac{5}{2} ( \dfrac{\pi}{2} - 0 + \dfrac{\pi}{8} ) = \displaystyle \left( \dfrac{5}{2} \right) \left( \dfrac{5 \pi}{8} \right) = \dfrac{25 \pi}{16}$

Therefore,

$G = \displaystyle \left( \dfrac{2}{\pi} \right) ( \dfrac{3 \pi}{8} , 0, \dfrac{25 \pi}{16} ) = ( \dfrac{3}{4} , 0, \dfrac{25}{8} )$

So that $x = \dfrac{3}{4} , y = 0, z = \dfrac{25}{8}$ , making the answer $4 x + 8 z = 3 + 25 = \boxed{28}$