\displaystyle\sum_{k=1}^{n}\frac{1}{n}f\Big(\frac{k}{n}\Big)-\int_{0}^{1}f(x)\,dx=\sum_{k=1}^{n}\int_{(k-1)/n}^{k/n}\Big(f\Big(\frac{k}{n}\Big)-f(x)\Big)\,dx