  In this note we introduce the notion of Newton--C\^{o}tes functionals
corrected by L\'{e}vy areas, which enables us to consider integrals of the type
$\int f(y) \mathrm{d}x,$ where $f$ is a ${\mathscr{C}}^{2m}$ function and $x,y$
are real H\"{o}lderian functions with index $\alpha>1/(2m+1)$ for all $m\in
{\mathbb{N}}^*.$ We show that this concept extends the Newton--C\^{o}tes
functional introduced in Gradinaru et al., to a larger class of integrands.
Then we give a theorem of existence and uniqueness for differential equations
driven by $x$, interpreted using the symmetric Russo--Vallois integral.
