= Solution
Height above the surface is $h=r-R$. Integrate the <density> over all exterior spherical shells:
$$
M=\int_R^\infty\int_0^\pi\int_0^{2\pi}
\rho_0e^{-(r-R)/h_0}r^2\sin\theta\,d\phi\,d\theta\,dr
=4\pi\rho_0\int_0^\infty(R+h)^2e^{-h/h_0}\,dh.
$$
Repeated <integration by parts>, or the substitution $h=h_0s$, gives
$$
\int_0^\infty e^{-h/h_0}\,dh=h_0,\quad
\int_0^\infty he^{-h/h_0}\,dh=h_0^2,\quad
\int_0^\infty h^2e^{-h/h_0}\,dh=2h_0^3.
$$
Expand $(R+h)^2$ and use these moments to obtain the <mass of an exponential spherical atmosphere>:
$$
\boxed{M=4\pi\rho_0h_0(R^2+2Rh_0+2h_0^2)}.
$$
For $h_0\ll R$, the leading term is surface area $4\pi R^2$ times column mass $\rho_0h_0$; the other terms retain the finite-curvature corrections.
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