Finding the mass using sphere coordinates

Okay! For further study you can also refer to

http://mathworld.wolfram.com/SphericalCoordinates.html

but for how we wanna solve the question in spherical coordinates. First we need to convert any thing to spherical equivalent for example converting $x^2+y^2+z^2=2z$ in spherical coordinates considering $x^2+y^2+z^2=R^2$ and $z=R\cos\theta$ is: $$R^2=2R\cos\theta\to R=2\cos\theta$$ also the hemisphere is $R=2$ where $z>0$ results in $\theta<\frac{\pi}{2}$ and the density function is $p(R,\theta,\phi)=R^2\cos\theta$. Now let's get to corresponding integral due to the problem. Before going to write it note that $dxdydz=R^2\sin\theta d\theta d \phi dR$: $$I=\iiint{p(R,\theta,\phi)}R^2\sin\theta d\theta d \phi dR=\int_{0}^{2\pi}\int_{0}^{{\pi\over 2}}\int_{2\cos\theta}^{2}R^4\sin\theta\cos\theta dRd\theta d\phi$$ Now the integral is simple to follow: $$I=2\pi\int_{0}^{{\pi\over 2}}\int_{2\cos\theta}^{2}R^4\sin\theta\cos\theta dRd\theta=2\pi\int_{0}^{{\pi\over 2}}\frac{32}{5}(1-\cos^5\theta)\sin\theta\cos\theta d\theta$$$$=\frac{64\pi}{5}\int_{0}^{1}u-u^6du=\frac{32\pi}{7}$$