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Math Help - Two questions from calc II

  1. #1
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    Two questions from calc II

    hey guys, thanks to those who helped me on my last problem, this time i've ufortunately got two

    1) for the integral x
    trap(10)= 12.676
    trap(30)= 10.420

    i have to find the actual value

    the error is n^2 when using trap so 20^2= 400, 12.676-10.420 = 2.256

    so is x = 2.256/400?

    i'm not really sure how to go about this



    2) this problem i don't really have any idea what to do whatsoever, and it's a proof (which are gods way of punishing those you take math classes)

    i need to integrate R(x) = t^(x-1)e^(-t)dt from zero to infinity

    then prove that R(x+1) = nRn

    i tried working out the anitderivate for the start, and i came out with just (t^(x+1)e^(-t)) / ln(t)

    i'm not really even sure if that is correct

    thanks in advance to anyone who can help me
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  2. #2
    hpe
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    Quote Originally Posted by msm1593
    hey guys, thanks to those who helped me on my last problem, this time i've ufortunately got two

    1) for the integral x
    trap(10)= 12.676
    trap(30)= 10.420

    i have to find the actual value
    the error is n^2
    The error is actually of the form \frac{C}{n^2} + O(n^{-4}), where  C depends on the second derivative of f. Let's assume it is exactly \frac{C}{n^2} , with unknown  C, and call the unknown integral I. Then you have the equations
     12.676 = I + \frac{C}{100}
    10.420 = I + \frac{C}{900}
    Thus your can solve for I: I = \frac{9 \cdot 10.420 - 12.676}{8} = 10.138 with an error of the order n^{-4} \sim 10^{-4}, thus correct to two or three decimal digits.
    This is called "Aitken extrapolation" and in this context as "Romberg integration", so you should give proper attribution.
    2) this problem i don't really have any idea what to do whatsoever, and it's a proof (which are gods way of punishing those you take math classes)

    i need to integrate R(x) = t^(x-1)e^(-t)dt from zero to infinity

    then prove that R(x+1) = nRn

    i tried working out the anitderivate for the start, and i came out with just (t^(x+1)e^(-t)) / ln(t)

    i'm not really even sure if that is correct
    No it isn't
    Try integration by parts in the improper integral. This is known as the Gamma function,
    \Gamma(x) = \int_0^\infty  t^{x-1}e^{-t}dt
    and the formula \Gamma(x+1) = x \Gamma(x) is the functional equation of the Gamma function.
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  3. #3
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    Quote Originally Posted by msm1593
    2) this problem i don't really have any idea what to do whatsoever, and it's a proof (which are gods way of punishing those you take math classes)

    i need to integrate R(x) = t^(x-1)e^(-t)dt from zero to infinity

    then prove that R(x+1) = nRn

    i tried working out the anitderivate for the start, and i came out with just (t^(x+1)e^(-t)) / ln(t)

    i'm not really even sure if that is correct

    thanks in advance to anyone who can help me
    Substitute x+1 into the equation you were given and integrate from there. It's not that bad.
    Last edited by Jameson; October 26th 2005 at 06:07 PM.
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