Results 1 to 7 of 7

Math Help - Help with Riemann sums for a continuous function

  1. #1
    Junior Member
    Joined
    Aug 2006
    Posts
    30

    Help with Riemann sums for a continuous function

    I couldn't get LaTEX to display this one, it kept erroring out after about 3/4 of the equation was entered. So I pasted a jpg instead. Thanks for looking.
    Attached Thumbnails Attached Thumbnails Help with Riemann sums for a continuous function-image1.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    I think there is a mistake. That is not expressable as a Riemann sum.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Aug 2006
    Posts
    30
    That's not good, I double-checked it and it is as typed in the course guide!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Sorry forgive me. I saw something else in the problem and made a mistake.
    The function is,
    \sin \pi x
    Then, the convergent riemann sum is, (by definition)
    \int_0^1 \sin \pi x dx

    Now,
    f(x)=-\frac{1}{\pi} \cos \pi x
    Has property that,
    f'(x)=\sin \pi x
    Then by the Fundamental Theorem of Calculus (eventhough I am angry when it is referred to as like this. It certainly is not a fundamental theorem at all!)
    -\frac{1}{\pi} \cos \pi +\frac{1}{\pi} \cos 0=2\pi
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Oct 2005
    From
    Earth
    Posts
    1,599
    Rewrite the sum like so.

    \lim_{n \rightarrow \infty}\sum_{i=1}^{n} \frac{\sin(\frac{\pi{i}}{n})}{n}

    Now let x=\frac{i}{n} and dx=\frac{1}{n}. Now all of PH's work follows.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Aug 2006
    Posts
    30
    The only example from my book that I have to go by is like the first image below. I tried to rewrite your work in the second image to anwser the question. Can you look at it and let me know if I messed up? Thanks
    Attached Thumbnails Attached Thumbnails Help with Riemann sums for a continuous function-image1.jpg   Help with Riemann sums for a continuous function-image2.jpg  
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Oct 2005
    From
    Earth
    Posts
    1,599
    You must first write the sum in a general way as I did for you. You are setting up rectangles partitioned \frac{1}{n},\frac{2}{n}...etc and letting n approach infinity for the rectangles height, and the width of each rectangle is \frac{1}{n}. Now when you apply the limit, the infinite sum tranforms into the integral \int_{0}^{1}\sin(\pi{x})dx.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: September 2nd 2010, 11:28 AM
  2. Riemann sums of a function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 22nd 2010, 12:22 PM
  3. Riemann sums
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 4th 2009, 11:26 AM
  4. Riemann Sums
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 11th 2008, 12:32 PM
  5. Riemann Sums and Riemann Integrals
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 1st 2006, 02:08 PM

Search Tags


/mathhelpforum @mathhelpforum