Question Number 66253 by mathmax by abdo last updated on 11/Aug/19
![prove by Rieman sum that ∫_0 ^1 xdx =(1/2)](https://www.tinkutara.com/question/Q66253.png)
$${prove}\:{by}\:{Rieman}\:{sum}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{xdx}\:=\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by mathmax by abdo last updated on 12/Aug/19
![∫_0 ^1 xdx =lim_(n→+∞) (1/n)Σ_(k=1) ^n (k/n) =lim_(n→+∞) (1/n^2 )Σ_(k=1) ^n k =lim_(n→+∞) (1/n^2 )((n(n+1))/2) =lim_(n→+∞) ((n^2 +n)/(2n^2 )) =lim_(n→+∞) (n^2 /(2n^2 )) =(1/2)](https://www.tinkutara.com/question/Q66298.png)
$$\int_{\mathrm{0}} ^{\mathrm{1}} {xdx}\:={lim}_{{n}\rightarrow+\infty} \frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{k}}{{n}}\:={lim}_{{n}\rightarrow+\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\sum_{{k}=\mathrm{1}} ^{{n}} \:{k} \\ $$$$={lim}_{{n}\rightarrow+\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:={lim}_{{n}\rightarrow+\infty} \:\frac{{n}^{\mathrm{2}} \:+{n}}{\mathrm{2}{n}^{\mathrm{2}} }\:={lim}_{{n}\rightarrow+\infty} \frac{{n}^{\mathrm{2}} }{\mathrm{2}{n}^{\mathrm{2}} }\:=\frac{\mathrm{1}}{\mathrm{2}} \\ $$