A few days ago I asked this question. I have solved what I was asking there. The mass of the given object is $$M=\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{\pi/3}\displaystyle\int_{0}^{1}k\displaystyle\frac{r}{3}cos(\theta)\displaystyle\frac{r^2}{6}sin(\theta)drd\theta d\phi$$$$+\displaystyle\int_{0}^{2\pi}\displaystyle\int_{\pi/3}^{\pi/2}\displaystyle\int_{0}^{2cos\theta}k\displaystyle\frac{r}{3}cos(\theta)\displaystyle\frac{r^2}{6}sin(\theta)drd\theta d\phi=\displaystyle\frac{5\pi k}{432}$$
Is it correct that, if the center of mass is $(\bar{x},\bar{y},\bar{z})$, $z=\frac{r}{3}cos(\theta)$ and the Jacobian of the transformation is $\frac{r^2sin^2(\theta)}{6}$ $$\bar{z}=\displaystyle\frac{1}{M}\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{\pi/3}\displaystyle\int_{0}^{1}\displaystyle\frac{r}{3}cos(\theta)\displaystyle\frac{r^2}{6}sin(\theta)drd\theta d\phi+\displaystyle\int_{0}^{2\pi}\displaystyle\int_{\pi/3}^{\pi/2}\displaystyle\int_{0}^{2cos\theta}\displaystyle\frac{r}{3}cos(\theta)\displaystyle\frac{r^2}{6}sin(\theta)drd\theta d\phi$$$$=\frac{1}{k}$$ This don't make sense to me, since the density of mass of the object is directly proportional to the plane $xy$, shouldn't the z-axis of the center of mass be directly proportional to $k$?