  We use nonstandard analysis to study the problem of expressing a Gaussian
integral in terms of the limiting behavior of a sequence of spherical
integrals. Peterson and Sengupta proved that if a Gaussian measure $\mu$ has
full support on a finite-dimensional Euclidean space, then the expected value
of a bounded measurable function on that domain can be expressed as a limit of
integrals over spheres $S^{n-1}(\sqrt{n})$ intersected with certain affine
subspaces of $\mathbb{R}^n$. This allows one to realize the Gaussian Radon
transform of such functions as a limit of spherical integrals. Using
nonstandard analysis, we study such limits in terms of Loeb integrals over a
single hyperfinite dimensional sphere. This nonstandard geometric approach
generalizes the known limiting result for bounded continuous functions to the
case when the Gaussian measure is not necessarily fully supported. We also
present an asymptotic linear algebra result needed in the above proof.
