  In this paper, we first derive a quantitative quermassintegral inequality for
nearly spherical sets in $\mathbb{H}^{n+1}$ and $\mathbb{S}^{n+1}$, which is a
generalization of the quantitative Alexandrov-Fenchel inequality proved in
$\mathbb{R}^{n+1}$ [22]. Then we use this method to derive the stability of
some geometric inequalities involving weighted curvature integrals and
quermassintegrals for nearly spherical sets in $\mathbb{R}^{n+1}$ and
$\mathbb{H}^{n+1}$.
