Integral of small function over small interval

Let $X=[0,1]$. For all $\varepsilon>0$, let $A_\varepsilon\subset X$ satisfy $\mathrm{Leb}(A_\epsilon)=O(\varepsilon)$. Further, suppose there exists a family of functions $f_\varepsilon \in L^1(X)$ such that$$\int_X f_\varepsilon(x)\, d\mathrm{Leb}(x) = O(\varepsilon).$$I am trying to prove that$$\int_{A_\epsilon} f_\varepsilon(x)\, d\mathrm{Leb}(x)=O(\varepsilon^2).$$I thought that integrating a small function over a small interval would contribute to an even smaller error. Is this possible to prove? Or would one really need to show that $f_\epsilon(x)=O(\epsilon)$ in order to prove something like this.