  In a $\mathcal{N}=2$ superconformal gauge theory with matter hypermultiplets
transforming in the symmetric and anti-symmetric representations of SU($N$), we
study the integrated correlators of two Coulomb-branch operators and two
moment-map operators using localization. In the corresponding matrix model we
identify the operator associated with the integrated insertions of moment-map
operators and provide for it an exact expression valid for all values of the
coupling constant in the planar limit. This allows us to study the integrated
correlators at strong-coupling where we show that they behave as the 2-point
functions of the Coulomb-branch operators, up to an overall constant dependent
only on the conformal dimensions of the latter. The strong-coupling relation
between integrated correlators and 2-point functions turns out to be the same
as in $\mathcal{N}=4$ SYM at large $N$, despite the reduced amount of
supersymmetry in our theory.
