prove-by-Rieman-sum-that-0-1-xdx-1-2-

Question Number 66253 by mathmax by abdo last updated on 11/Aug/19

p r o v e b y R i e m a n s u m t h a t \int_{0}^{1} x d x = \frac{1}{2}

Commented by mathmax by abdo last updated on 12/Aug/19

\int_{0}^{1} x d x = {l i m}_{n \to + \infty} \frac{1}{n} \sum_{k = 1}^{n} \frac{k}{n} = {l i m}_{n \to + \infty} \frac{1}{n^{2}} \sum_{k = 1}^{n} k

= {l i m}_{n \to + \infty} \frac{1}{n^{2}} \frac{n (n + 1)}{2} = {l i m}_{n \to + \infty} \frac{n^{2} + n}{2 n^{2}} = {l i m}_{n \to + \infty} \frac{n^{2}}{2 n^{2}} = \frac{1}{2}