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Math Help - Express Integral as a Riemann Sum and solve

  1. #1
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    Express Integral as a Riemann Sum and solve

    Hello MHF,
    I have been able to process this question a little bit, but got a bit stuck when i got to a certain point..

    Express the integral as a limit of sums;
    \int_{0}^{\pi}\sin5x \ dx

    So by setting \Delta x=\frac{\pi}{n} and x_{i}=\frac{\pi i}{n}
    Then, \lim_{n\to\infty}\sum_{i=1}^{n}\sin(\frac{5\pi  i}{n})*\frac{\pi}{n}

    \lim_{n\to\infty}\frac{\pi}{n}\sum_{i=1}^{n}\sin(\  frac{5\pi i}{n})<br />
    \lim_{n\to\infty}\frac{\pi}{n}\sin(\frac{5\pi}{n}\  sum_{i=1}^{n}i)<br />
    \lim_{n\to\infty}\frac{\pi}{n}\sin(\frac{5\pi}{n}*  \frac{n^2+n}{2})<br />
    \lim_{n\to\infty}\frac{\pi}{n}\sin(\frac{5\pi}{2}*  (1+n))<br />

    But form here I dont know if i have done something incorrectly, or how to keep reducing it to get rid of n..
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  2. #2
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    Quote Originally Posted by Robb View Post
    Express the integral as a limit of sums;
    \int_{0}^{\pi}\sin5x \ dx
    So by setting \Delta x=\frac{\pi}{n} and x_{i}=\frac{\pi i}{n}
    Then, \lim_{n\to\infty}\sum_{i=1}^{n}\sin(\frac{5\pi  i}{n})*\frac{\pi}{n}

    \lim_{n\to\infty}\frac{\pi}{n}\sum_{i=1}^{n}\sin(\  frac{5\pi i}{n})<br />
    \color{red}\lim_{n\to\infty}\frac{\pi}{n}\sin(\fra  c{5\pi}{n}\sum_{i=1}^{n}i)<br />
    The part is red above is incorrect. The sine function is not additive.
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  3. #3
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    Ah yes, that makes sense. Thanks allot for that.
    I actually just re-read the question, and it says to use a computer to calculate the sum and the limit... Is there away to analyse trigonometric functions using riemann sums without a computer for exam preperation(ie. can this question be taken any further)?

    Thanks.
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