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Math Help - Two problems involving FDC/Riemman sums

  1. #1
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    Two problems involving FDC/Riemman sums

    I have two questions I need help with. I have attempted both of them but I have no idea how to proceed from here.

    Question 1:
    Two problems involving FDC/Riemman sums-codecogseqn.gif
    Attempted solution:
    Two problems involving FDC/Riemman sums-img_0451.jpg
    I used product rule then I solved for d/dx of the integral but I have no idea how to solve for the integral in the first term of my final equation.

    Question 2:
    Two problems involving FDC/Riemman sums-codecogseqn-2-.gif

    Attempted solution:
    Two problems involving FDC/Riemman sums-img_0452.jpg
    The question gave a hint: "Rewrite it as lim_n-> infinity 1/n(*) then relate what you get to a Riemann sum. I got it to the point where I got the 1/n term out of the sum but I have no idea what the formula for the sum of a square root is and I can't find it anywhere.

    I'm not looking for the full solution I just want to know if I did so far is correct and how to continue further.
    Thanks in advance
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  2. #2
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    Re: Two problems involving FDC/Riemman sums

    For the first one, you have the average value of the function f(t) = \sqrt{1+t^3} over the interval (0,x^2). But, there is not really any simplification you can do.

    For the second one, a Riemann sum looks like this:

    \lim_{n\to \infty} \sum_{i = 1}^n f(x_i^*)\Delta x

    In general, \Delta x = \dfrac{b - a}{n} and x_i = a + i\Delta x.

    So, instead of factoring out \sqrt{\dfrac{1}{n^3}}, you should only factor out \Delta x. So, you want \sqrt{\dfrac{i}{n^3}} = \left(\sqrt{a + i\Delta x}\right)\Delta x. The hint is to factor out \dfrac{1}{n}. So, that must be \Delta x:

    \Delta x = \dfrac{b-a}{n} = \dfrac{1}{n}

    That means b-a = 1. Since the square root does not have a sum inside, it appears that a=0, b=1. Can you figure out the rest?
    Thanks from nubshat
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  3. #3
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    Re: Two problems involving FDC/Riemman sums

    I got it now thanks alot
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