Could somebody please explain how: $$\lim_{n\to\infty}\frac1n\sum_{r=0}^{n}\left(\frac{r}{n}\right)=\int_0^1x\,dx$$ I am just beginning integration and this has been highlighted as a very fundamental and important result/theorem in my senior's notes. Does it have any name ? Any books you could suggest would also be very helpful.
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2This is an application of Riemann sums/the fundamental theorem of calculus. – abnry May 22 '14 at 13:27
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Could you please elaborate. – User5617 May 22 '14 at 13:28
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1@User5617: http://en.wikipedia.org/wiki/Riemann_integral#Riemann_sums – Clement C. May 22 '14 at 13:35
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I don't have time and/or care to elaborate. Have you read your textbook carefully? – abnry May 22 '14 at 13:54
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1For calculus books, this question might be useful...http://math.stackexchange.com/questions/322892/calculus-book-recommendations-for-complete-beginner – Indrayudh Roy May 22 '14 at 13:56
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Firstly, notice that $x \mapsto x$ is a continuous function on $\mathbb{R}$, and hence uniformly continuous on the closed interval $[0,1]$, thus its Riemann integral on $[0,1]$ exists, and hence can be expressed as a limit of the left/right Riemann sum using partitions with strips of width $\frac{1}{n}$ for integer $n$. Secondly, the expression inside your limit on the left-hand side is just zero plus the right Riemann sum.
user21820
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The $\text{LHS}$ is area under $y=x$ from $x=0$ to $x=1$.
It divides your triangle in infinitesimal thick rectangles with width $\frac1n$ and the term $\frac r n$ divides $x=0\to1$ as height of each rectangle.
$\text{RHS}$ is also area according to fundamental theorem of calculus.
evil999man
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