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Math Help - Integration by limit of sum

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    Integration by limit of sum

    Hello...

    I need to calculate this integral using the limit of the summation.

    The period [0,T] is split into N intervals and the value point is at the left ot the interval.

    <br />
\int_0^{T} x^3 (t)\ dX(t)\ = \lim_{N\to \infty}\sum_{i=0}^{N-1} X_{i}^{3}(X_{i+1} - X_{i})<br />

    Can someone please help?

    Many Thanks to Galactus for his yesterday's idea but I believe my problem was not well posed.

    I know the solution for something very similar:

    <br />
\int_0^{T} 2X(t)\ dX(t)\ = \lim_{N\to \infty}\sum_{i=0}^{N-1} 2X_i (X_{i+1} - X_{i}) = X^2(T) - T<br />

    This is verifiable using the trick 2ab= (a+b)^2 -a^2 -b^2 and making a=X(i), b=X(i+1)-X(i)
    Straight, basic telescopic series come out and give the result.
    Last edited by paolopiace; March 4th 2008 at 11:08 AM. Reason: Telescopic series
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