Question

Prove that : ${\int }_0^{p}f\left(x\right)dx=\underset{n\to \infty }{lim}\frac{1}{n}\sum _{i=1}^{np}f\left(\frac{i}{n}\right)$

Answer 1

RHS: ${lim}_{n\to \infty }\frac{1}{n}{\sum }_{i=1}^{np}f\left(\frac{i}{n}\right)$

Let $\frac{i}{n}=x$ and $\frac{1}{n}=dx$

When $i=1,x=\frac{1}{n}=\frac{1}{\alpha }=0$

When $i=np,x=\frac{np}{n}=p$

$\therefore {lim}_{n\to \infty }\frac{1}{n}{\sum }_{i=1}^{np}f\left(\frac{i}{n}\right)={\int }_0^{p}f\left(x\right)dx$ = LHS