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

prove by Rieman sum that  ∫_0 ^1  xdx =(1/2)

$${prove}\:{by}\:{Rieman}\:{sum}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{xdx}\:=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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)

$$\int_{\mathrm{0}} ^{\mathrm{1}} {xdx}\:={lim}_{{n}\rightarrow+\infty} \frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{k}}{{n}}\:={lim}_{{n}\rightarrow+\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\sum_{{k}=\mathrm{1}} ^{{n}} \:{k} \\ $$$$={lim}_{{n}\rightarrow+\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:={lim}_{{n}\rightarrow+\infty} \:\frac{{n}^{\mathrm{2}} \:+{n}}{\mathrm{2}{n}^{\mathrm{2}} }\:={lim}_{{n}\rightarrow+\infty} \frac{{n}^{\mathrm{2}} }{\mathrm{2}{n}^{\mathrm{2}} }\:=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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