I am new here. My problem: There is an integral $I:=\int_0^1 f(x)\,dx$ for $f\colon [0,1]\to\mathbb{R}$ and I want to compute it by $H_n:=\frac{f\left(0\right)}{2n}+\frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\dotsm + f\left(\frac{n-1}{n}\right)\right]+\frac{f\left(1\right)}{2n}$
Whats an easy way to prove the error $|I-H_n|\leq \frac{L}{4n}$?
The function $f$ suffices the lipsch. condition $\Vert f(x_1)-f(x_2) \Vert\leq L \vert x_1-x_2\Vert$?
I didn't attend any lecture about numerical analysis yet. I know how it looks like. I take equidistant steps an only compute the mean of these values. From the lip. condition I know that $\max_{x\in [0,1]} \frac{d}{dx}f(x)\leq L$.
Is there some literature or an EASY way to see this inequality? I think induction in $n$ won't make sense. I don't even know if this approximation has a name in the literature.