Let $f:[0,1]\rightarrow[0,1]$ be defined by
$$f(x)=\frac{2^{k}-1}{2^{k}}$$ for $$x\in[\frac{2^{k+1}-1}{2^{k+1}},\frac{2^{k}-1}{2^{k}}],$$ $k\geq1$. Then $f$ is a Riemann integrable function such that
$$\int_{0}^{1}f(x)dx=1$$
$$\int_{0}^{1}f(x)dx=\frac{2}{3}$$
$$\frac{1}{2}<\int_{0}^{1}f(x)dx<\frac{2}{3}$$