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I am to compute Compute $\int_0^2 x^3 d[x]$, where $[x]$ is the greatest integer function. I can think of this somewhat graphically, where at $[0,1)$ a straight line lies at $y=0$, $[1,2)$ a straight line lies at $y=1$. But using the Riemann Stieltjes integral, I am somewhat confused. Could anyone help?

Thanks

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1 Answer 1

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$$\int_0^2 x^3\,d[x]=\sum_{n=1}^2n^3=9$$

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    $\begingroup$ Could you please elaborate why the integral becomes the sum? I don't see it right away. I tried to divide the integral to sub-intervals of the form $[n,n+1)$ and say that $[x]$ is the same as $x$ on these sub-intervals but I got a different answer. In fact, I got $4$. I would appreciate it if you told me what I had done wrong as well. Thanks. $\endgroup$
    – stressed out
    Nov 29, 2017 at 15:50
  • $\begingroup$ It has long story so I prefer to refer you to this: math.stackexchange.com/a/2142052/108128 $\endgroup$
    – Nosrati
    Nov 29, 2017 at 15:55
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    $\begingroup$ OK. Thanks. But I think you need to mention that in your answer, because it isn't something obvious at all. $\endgroup$
    – stressed out
    Nov 29, 2017 at 15:58
  • $\begingroup$ Some more explanation would work $\endgroup$
    – user402326
    Nov 29, 2017 at 18:57
  • $\begingroup$ @sadlyfe It's important that in interval $(n,n+1)$ the varieties of $[x]$ is $0$, that is $d[x]=0$. Thus the integral evaluate in jump points, they are $n$. $\endgroup$
    – Nosrati
    Nov 29, 2017 at 19:24

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