In this note we introduce the notion of Newton–Côtes functionals corrected by Lévy areas, which enables us to consider integrals of the type ∫ f(y) dx, where f is a C2m function and x, y are real Hölderian functions with index α>1/(2m+1) for all m∈ℕ*. We show that this concept extends the Newton–Côtes 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.