When viewing the radius of gyration of a sphere, and trying to verify my own calculations $R_g^2=\frac25R^2$ against the online material , while the later shows $R_g^2=\frac35R^2$ and does not use the squared moment arm $r\sin\theta$ if rotating about $z-$axis.
EDIT
The derivation should br trivial in spherical coordinate system with the volume element $\mathrm{d}V=r^2\sin\theta\mathrm{d}r\mathrm{d}\theta\mathrm{d}\varphi$. The moment arm shall be $r\sin\theta$. Assuming that the sphere is homogeneous in density ($\rho$ is constant), therefore, the square of the radius of gyration about the axis through the sphere center is expressed as the moment of inertia divided by the mass, i.e., \begin{align} R_g^2&=\frac{\rho\int_0^Rr\mathrm{d}r\int_0^{2\pi}\mathrm{d}\varphi\int_0^{\pi}r\sin\theta\mathrm{d}\theta(r\sin\theta)^2}{\rho\frac43\pi R^3}\\ &=\frac{2\pi\frac{R^5}{5}\frac43}{\frac43\pi R^3}\\ &=\frac25 R^2. \end{align}
where $\frac43$ in the nominator is from $\int\sin^3(\theta)\mathrm{d}\theta=\frac{\cos3\theta}{12} - \frac{3\cos\theta}{4} + c$.
Requested from Comments that I should present my own calculations.
I checked Wiki which shows that "Solid sphere (ball) of radius r and mass m" has exactly $\frac25R^2$. I was puzzled and wondered to write to the author, but no contact information can be found. What tricks are hidden inside his derivation or some common definition rule I am not aware of?