I. INTRODUCTION Quantum fluctuations of surfaces are important in many physical phenomena. It is less known these fluctuations can sometimes be expressed in term of a functional integral over reparametrizations of variables relevant in the Feynman path integral. In this paper we shall consider the expression where x ( θ ) is the boundary curve C and where The functional (2) is known in mathematics as the Douglas integral [1], whose minimum with respect to variations of the reparametrizations θ gives the minimal area Here θ ∗ ( σ ) is the saddle point of the integral (1). The functional integration (1) thus gives the area law to leading order. The path integral over reparametrizations (1) was introduced in this context in Ref. [2] in connection with an off-shell string propagator. More recently it was proposed by Polyakov [3] as an ansatz for the Wilson loop in large N QCD. The leading behavior obviously gives the leading behavior of the Wilson loop, found in most string models, where the bulk field X $^{µ}$( τ, σ ) satisfies the Dirichlet boundary condition To derive Eq. (3) from the Nambu-Goto action one can follow Douglas [1], or the more recent elegant paper by Migdal [4]. The functional integral (1) can be expanded around the saddle point θ ∗ . With θ ( σ ) = θ ∗ ( σ ) + β ( σ ) we obtain the first non-vanishing contribution Here σ ∗ ( θ ) is the inverse function of θ ∗ ( σ ). Some dynamical consequences of Eq. (5) have been discussed by Rychkov [5] and by the authors [6]. However, it is perhaps fair to say that the physical meaning of the fluctuation integral (5) is not so clear. X $^{µ}$( τ, σ ) | boundary = x $^{µ}$( σ ) . (4) A$_{2}$ = K 8 π ∫ 2 π 0 d θ ∫ 2 π 0 d θ ′ ˙ x ( θ ) · ˙ x ( θ ′ ) ( β ( θ ) − β ( θ ′ )) 2 1 − cos( σ ∗ ( θ ) − σ ∗ ( θ ′ )) (5) 2 A [ x ( θ ∗ )] = S$_{min}$ ( C ) . (3) A [ x ( θ )] = 1 8 π ∫ 2 π 0 d σ ∫ 2 π 0 d σ ′ ( x ( θ ( σ )) − x ( θ ( σ ′ ))) 2 1 − cos( σ − σ ′ ) . (2) W ( C ) ≡ ∫ D θ e − KA [ x ( θ $^{)]}$, (1)