How to approximate integrals of a function by a function of integrals?

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Consider the following integral over the whole space

$$I = \int \sqrt{h^{ij}(x) A_i(x) A_j(x)} d^3 x$$
where $A_i$ are the components of a highly localized vector field, and $h^{ij}$ is some well behaved Riemannian metric. One can imagine the magnitude of the components of the vector field to scale roughly with a normalized spatial Gaussian with a variance length $\sigma$. In the limit $\sigma \to 0$, or when the vector field gets localized to almost a delta peak around a certain point $x_0$, we can write an approximation to the integral as

$$I \approx \sqrt{ h^{ij}(x_0) \int A_i (x) d^3 x \int A_j(y) d^3 y }$$

I believe there are similar higher order corrections to this expression when the field becomes very slightly non-localized. These would involve terms such as $\int (x^j - x^j_0) A_i dx, \int (x^j - x^j_0)(x^k - x^k_0) A_i dx, ...$ (without having to refer to $\sigma$ or anything similar) and the metric structure evaluated only at $x_0$. However, I do not know how to methodically search for these corrections.

In other words, I am looking for an approximation scheme which would express the integral $I$ in terms of functionals of $A_i(x)$, its derivatives and the coordinates, while only referring to $h^{ij}$ and its derivatives at some point $x_0$ which is chosen so that it is very close to some centroid of the $A_i$s.

How can I do that?

(I have posted a related version of this problem at Math Stackexchange.)

The correction terms would have order $O(\sigma)$. But by expanding the metric around zero one sees already that one gets at first order a term depending on the derivative of the metric at $x_0$. Thus the metric at $x_0$ alone is not sufficient to give better accuracy.
@ArnoldNeumaier I agree, and this is also mentioned in the question. Of course, when the dust settles, I believe the expansion should even be possible to state in terms of momenta of the vector field computed with respect to derivatives of the geodesic exponential around $x_0$, and the expansion would then refer only to curvature and other covariant geometric tensors at $x_0$.
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