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Question Number 168799 by MikeH last updated on 17/Apr/22
If the function f is continuous in  [a,b]   prove that    lim_(x→∞ ) ((b−a)/n)Σ_(k=1) ^n f(a+((k(b−a))/n))=∫_a ^b f(x)dx
$$\mathrm{If}\:\mathrm{the}\:\mathrm{function}\:{f}\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{in} \\ $$$$\left[{a},{b}\right]\: \\ $$$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$\:\underset{{x}\rightarrow\infty\:} {\mathrm{lim}}\frac{{b}−{a}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{f}\left({a}+\frac{{k}\left({b}−{a}\right)}{{n}}\right)=\int_{{a}} ^{{b}} {f}\left({x}\right){dx} \\ $$

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