  Given a compact planar region $A$, let $\tau_A$ be the (random) time it takes
for $A$ to be fully covered by a spatial birth-growth process in $A$ with seeds
arriving as a unit-intensity Poisson point process in $A \times [0,\infty)$,
where upon arrival each seed grows at unit rate in all directions. We show that
if $\partial A$ is smooth or polygonal then $\Pr [ \pi \tau_{sA}^3 - 6 \log s -
4 \log \log s \leq x]$ tends to $\exp(- (\frac{81}{4\pi})^{1/3} |A|e^{-x/3} -
  (\frac{9}{2\pi^2})^{1/3} |\partial A| e^{-x/6})$ in the large-$s$ limit; the
second term in the exponent is due to boundary effects, the importance of which
was not recognized in earlier work on this model. We present similar results in
higher dimensions (where boundary effects dominate). These results are derived
using new results on the asymptotic probability of covering $A$ with a
high-intensity spherical Poisson Boolean model \emph{restricted to $A$} with
grains having iid small random radii, which generalize recent work of the
author that dealt only with grains of deterministic radius.
