= Solution
Use <spherical coordinates> $x=s\sin\theta\cos\phi$, $y=s\sin\theta\sin\phi$, $z=s\cos\theta$. The <density> becomes $\rho_0s\sin\theta$, and the <Jacobian determinant> supplies $dV=s^2\sin\theta\,ds\,d\theta\,d\phi$. Therefore the <mass> is the <volume integral>
$$
M=\rho_0\int_0^{2\pi}d\phi\int_0^\pi\sin^2\theta\,d\theta\int_0^1s^3\,ds.
$$
Using $\sin^2\theta=(1-\cos2\theta)/2$, the three <integrals> are $2\pi$, $\pi/2$ and $1/4$. Hence
$$
\boxed{M=\frac{\pi^2\rho_0}{4}}.
$$
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