= Solution
Put $D=d-1$ for the membrane's internal dimension and assume $\sigma>0$. The small-slope expansion is
$$
\sqrt{1+|\nabla h|^2}=1+\tfrac12|\nabla h|^2+O(|\nabla h|^4).
$$
The zeroth-order term is the projected area and is independent of the height field. Discarding it changes only an overall normalization of the <partition function>. Thus the leading fluctuation energy and its statistical weight are
$$
H_2[h]=\frac\sigma2\int d^Dx\,|\nabla h|^2,\qquad
\boxed{\mathcal Z_2=\int\mathcal Dh\,\exp\left[-\frac{\beta\sigma}{2}\int d^Dx\,|\nabla h|^2\right].}
$$
One obtains this <Gaussian field theory> by discretizing the height field on a microscopic mesh, integrating every height with its Boltzmann weight, and then writing the continuum measure. A microscopic cutoff is retained for quantities dominated by short wavelengths. Quadratic order here really means small slopes: a large rigid height shift is still free because the energy has no dependence on $h$ itself.
The uniform height mode has zero energy. Fix its mean, impose a <boundary condition>, or divide out the integral over global translations to normalize the <partition function>. Height differences are independent of that choice. The approximation is the tension-only part of <small-slope elastic membrane energy>; there is no bending rigidity or confining potential in this part.
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