The paper studies the asymptotics of the Brownian integrals with paths restricted to a bounded domain of $\Rn$, when the domain is dilated to infinity. The framework is that of the Bose-Einstein statistics with paths observed within random time intervals which are integer multiplies of some fixed $\beta>0$. The three first terms of the asymptotics are found explicitly via the functional integrals. In the case of a gas of interacting Brownian loops an expression for the volume term of the asymptotics of the log-partition function is found and the correction term is proved to by order be the boundary area of the domain.