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Math Help - Integrate, Riemann.

  1. #1
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    Integrate, Riemann.

    Hi, i can't solve this, please if someone can solve and show me the steps (:
    $\displaystyle \int_{1}^{e} \ln (x) dx $ P is the partition. $P:[q^{0};q^{1};q^{2};...;q^{n}]$ ; $\displaystyle q=e^{\frac{1}{n}}$


    Thanks for all the answers (:
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  2. #2
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    Re: Integrate, Riemann.

    Quote Originally Posted by SimpleMan View Post
    Hi, i can't solve this, please if someone can solve
    What do you mean by "solve" here?

    Quote Originally Posted by SimpleMan View Post
    $\displaystyle \int_{1}^{e} \ln (x) dx $ P is the partition. $P:[q^{0};q^{1};q^{2};...;q^{n}]$ ; $\displaystyle q=e^{\frac{1}{n}}$
    What do you mean by $[q^{0};q^{1};q^{2};...;q^{n}]$ ; $\displaystyle q=e^{\frac{1}{n}}$?
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  3. #3
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    Re: Integrate, Riemann.

    so let ln(x) be f-1(x) so ex=f(x)
    F-1(x)=xf-1(x)-F(f-1(x))
    so
    = x(ln(x)- eln(x) = xln(x)-x evaluated at e and 1

    which should be 1

    if you haven't seen the equation before look at the geometry between a function and its inverse... they add up to make a rectangle.

    I also don't know what the partition thingamabobs are but the answer is 1
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  4. #4
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    Re: Integrate, Riemann.

    Yes, that is easy to integrate but I suspect Simple Man is asking about the Riemann sum that will give that integral. The partition is e^{0/n}=1, e^{1/n}, e^{2/n}, ..., e^{n/n}= e. Those values for x make it easy to find the corresponding "f(x)": ln(1)= 0, ln(e^{1/n})= \frac{1}{n}, ln(e^{2/n})= \frac{2}{n}, ..., ln(e)= 1. However, it is also makes the interval lengths variable- the length of the first interval is e^{1/n}- 1, the second e^{2/n}- e^{1/n}, etc.

    The Riemann sum will be \frac{e^{2/n}- e^{1/n}}{n}+ \frac{2(e^{3/n}- e^{2/n})}{n}+ \frac{3(e^{4/n}- e^{3/n})}{n}+ \cdot\cdot\cdot+ \frac{n(e- e^{(n-1)/n}}{n} which is, of course, the same as
    \frac{1}{n}(ne- e^{(n-1)/n}- \cdot\cdot\cdot- e^{2/n}+ e^{1/n})
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