  Owing to their favorable scaling with dimensionality, Monte Carlo (MC)
methods have become the tool of choice for numerical integration across the
quantitative sciences. Almost invariably, efficient MC integration schemes are
strictly designed to compute ratios of integrals, their efficiency being
intimately tied to the degree of overlap between the given integrands.
Consequently, substantial user insight is required prior to the use of such
methods, either to mitigate the oft-encountered lack of overlap in ratio
computations, or to find closely related integrands of known quadrature in
absolute integral estimation. Here a simple physical idea--measuring the volume
of a container by filling it up with an ideal gas--is exploited to design a new
class of MC integration schemes that can yield efficient, absolute integral
estimates for a broad class of integrands with simple transition matrices as
input. The methods are particularly useful in cases where existing (importance
sampling) strategies are most demanding, namely when the integrands are
concentrated in relatively small and unknown regions of configuration space
(e.g. physical systems in ordered/low-temperature phases). Examples ranging
from a volume with infinite support to the partition function of the 2D Ising
model are provided to illustrate the application and scope of the methods.
