= Logarithmic integrals in a parabolic lubrication gap
For a fixed outer radius $b\ll a$ and $h=a\epsilon+r^2/(2a)$,
$$
\int_0^b\frac{r^{2n-1}}{h^n}dr=2^{n-1}a^n\log(1/\epsilon)+O(a^n),\qquad n\geq1.
$$
Indeed, for $a\sqrt\epsilon\ll r\ll b$ the integrand is $(2a)^n/r$; integrating from a scale proportional to $a\sqrt\epsilon$ to $b$ produces half of $\log(1/\epsilon)$. The inner region contributes a bounded term after rescaling $r=a\sqrt\epsilon\,s$. In particular, $\int h^{-1}dxdy$ and $\int x^2/(ah^2)dxdy$ have the same leading coefficient $2\pi a$.
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