Scalar curvature integrals on geodesic balls

We study scalar curvature integrals over geodesic balls on complete noncompact manifolds. Under nonnegative Ricci curvature and rotational symmetry, we prove the sharp bound $${\int }_{B_p(r)}S\text{d}{\mu }_g\le 2n(n-1){\omega }_n(1-v^{1\mathrm{/}(n-1)})r^{n-2},$$ when $p$ is the rotational origin. Here ${\omega }_n$ is the volume of the Euclidean unit ball and $v=V_{M^n}$ is the asymptotic volume ratio. In dimension three, with the center at a pole, we obtain a sharp bound $${\int }_{B_p(r)}S\text{d}{\mu }_g\le 16\pi r,$$ without any positive volume growth assumption. We also obtain a bound for every asymptotic volume ratio which is sharp for $0\le v\le 1\mathrm{/}4$. Under nonnegative sectional curvature and $v>0$, we prove the explicit bound at arbitrary centers $${\int }_{B_p(r)}S\text{d}{\mu }_g\le 12\pi (1-v)v^{-3}r\mathrm{.}$$ If $0\le 1-v\le 1\mathrm{/}100$, we further obtain $${\int }_{B_p(r)}S\text{d}{\mu }_g\le [8\pi (1-v)+200\pi (1-v{)}^2]r\mathrm{.}$$ The leading coefficient $8\pi$ is sharp, and a remainder of quadratic order in $1-v$ is necessary.