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Thread: integration theorem!

  1. #1
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    integration theorem!

    i need some help or understanding or a book or material that explains this type of problem

    find the value of

    $\displaystyle \int_{0.3}^{2.6} x^3 d(x + [x] ) $

    thanks
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  2. #2
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    Re: integration theorem!

    That is an example of a "Stieljes" integral. The original "Riemann" integral, $\displaystyle \int_a^b f(x)dx$, divides the x-axis into sub-intervals, $\displaystyle [x_i, x_{i+1}]$, and then measures the "area" by $\displaystyle f(x)\Delta x$ where $\displaystyle \Delta x$ is just the length, $\displaystyle x_{i+1}- x_i$. The "Stieljes" integral, $\displaystyle \int_a^b f(x)d\alpha(x)$, generalizes that by measuring each sub-interval by $\displaystyle \alpha(x_{i+1})- \alpha(x_i)$ where $\displaystyle \alpha(x)$ can be any increasing function of x. If $\displaystyle \alpha(x)$ happens to be differentiable the Stieljes integral reduces to a Riemann integral: $\displaystyle \int_a^b f(x)d\alpha= \int_a^b f(x)\alpha'(x)dx$.

    But $\displaystyle \alpha(x)$ does not have to be differentiable or even continuous. One application of the the Stieljes integral is to be able to write a sum as an integral: if $\displaystyle \alpha(x)= \lfloor x\rfloor$ then $\displaystyle \alpha(x_{i+1})- \alpha(x_i)$ is non-zero only for $\displaystyle x_{i+1}$ and $\displaystyle x_i$ to be on different sides of an integer: $\displaystyle \int_0^5 f(x)d\lfloor x\rfloor= f(0)+ f(1)+ f(2)+ f(3)+ f(4)+ f(5)$.

    That, together with the fact that
    $\displaystyle \int_{0.3}^{2.6} x^3d(x+ \lfloor x\rfloor)= \int_{0.3}^{2.6}x^3 dx+ \int_{0.3}^{2.6} x^3 d\lfloor x\rfloor$
    should be enough for you to do this problem.

    (By the way, I have interpreted your "[x]" as the floor function. Latex has "lfloor" and "rfloor" to get $\displaystyle \lfloor x\rfloor$.
    Last edited by HallsofIvy; Jun 29th 2013 at 06:34 AM.
    Thanks from lawochekel, Shakarri and topsquark
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  3. #3
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    Re: integration theorem!

    do u carry out normal integration on

    $\displaystyle \int_{0.3}^{2.6} x^3 dx $

    thanks
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  4. #4
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    Re: integration theorem!

    Yes, of course.
    Thanks from lawochekel
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