  We consider conformal perturbation theory for $n$-point functions on the
sphere in general 2D CFTs to first order in coupling constant. We regulate
perturbation integrals using canonical hard disk excisions of size $\epsilon$
around the fixed operator insertions, and identify the full set of counter
terms which are sufficient to regulate all such integrated $n$-point functions.
We further explore the integrated 4-point function which computes changes to
the structure constants of the theory. Using an $sl(2)$ map, the three fixed
locations of operators are mapped to $0$, $1$, and $\infty$. We show that
approximating the mapped excised regions to leading order in $\epsilon$ does
not lead to the same perturbative shift to the structure constant as the exact
in $\epsilon$ region. We explicitly compute the correction back to the exact in
$\epsilon$ region of integration in terms of the CFT data. We consider the
compact boson, and show that one must use the exact in $\epsilon$ region to
obtain agreement with the exact results for structure constants in this theory.
