  The circle method has been successfully used over the last century to study
rational points on hypersurfaces. More recently, a version of the method over
function fields, combined with spreading out techniques, has led to a range of
results about moduli spaces of rational curves on hypersurfaces. In this paper
a version of the circle method is implemented in the setting of the
Grothendieck ring of varieties. This allows us to approximate the classes of
these moduli spaces directly, without relying on point counting, and leads to a
deeper understanding of their geometry.
