  In this paper, we consider the extreme behavior of a Gaussian random field
$f(t)$ living on a compact set $T$. In particular, we are interested in tail
events associated with the integral $\int_Te^{f(t)}\,dt$. We construct a
(non-Gaussian) random field whose distribution can be explicitly stated. This
field approximates the conditional Gaussian random field $f$ (given that
$\int_Te^{f(t)}\,dt$ exceeds a large value) in total variation. Based on this
approximation, we show that the tail event of $\int_Te^{f(t)}\,dt$ is
asymptotically equivalent to the tail event of $\sup_T\gamma(t)$ where
$\gamma(t)$ is a Gaussian process and it is an affine function of $f(t)$ and
its derivative field. In addition to the asymptotic description of the
conditional field, we construct an efficient Monte Carlo estimator that runs in
polynomial time of $\log b$ to compute the probability
$P(\int_Te^{f(t)}\,dt>b)$ with a prescribed relative accuracy.
