Let $f:[a,b] \to \mathbb{R}$ be a differentiable function such that $f'$ is Riemann-integrable.
Let $A_n = \frac{b-a}{n} \cdot \sum\limits_{k=0}^{n-1} { f \left( a+k \frac{b-a}{n} \right)} \, , \forall \, n \in \mathbb{N}^* $. Prove that $ \lim\limits_{n \to \infty} n \left( A_n - \int_a^b f(x) \, \mathrm dx \right) = \frac{b-a}{2} \cdot [ f(a)-f(b) ]$.
My guess would be that we need to use that $\int_a^b f'(x) \, \mathrm dx = f(b)-f(a)$, but I don't have any other idea how to tackle this problem.
