Results 1 to 5 of 5

Math Help - Riemann integral

  1. #1
    Member
    Joined
    May 2008
    Posts
    87

    Riemann integral

    f(x) = 0 if 0 <= x < 1 or 1 < x <= 2
    f(x) = 1 for every other value of x

    The small and large f is the same function, I have no idea why the font size changes, sorry about that.

    Show that f is integrable on [0, 2] and find \int_0^2 \! f(x) \, dx

    This is an example from a calculus textbook, the solution is given below. I'm having a problem with understanding how the partition is made as in the solution provided it is simply stated that this is the partition without any reason given for how the partition was decided to be like that.

    My question is simply how was it determined that the partition should be as stated? I have no doubt that it is correct, I simply don't understand it.

    Solution:

    Let \epsilon > 0 be given.
    Let P = {0, 1 - \epsilon/3, 1 + \epsilon/3, 2}

    Then L(f, P) = 0, since f(x) = 0 at points of each of these subintervals into which P subdivides [0, 2].

    Since f(1) = 1, we have:

    U(f, P) = 0(1 - \epsilon/3) + 1(2 \epsilon/3) + 0(2 - (1 + \epsilon/3))
    U(f, P) = (2 \epsilon)/3

    Hence, U(f, P) - L(f, P) < \epsilon and f is integrable on [0, 2]. Since L(f, P) = 0 for every partition,
    \int_0^2 \! f(x) \, dx = I_* = 0.


    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,922
    Thanks
    1762
    Awards
    1
    Quote Originally Posted by posix_memalign View Post
    f(x) = 0 if 0 <= x < 1 or 1 < x <= 2
    f(x) = 1 for every other value of x
    Show that f is integrable on [0, 2] and find \int_0^2 \! f(x) \, dx
    My question is simply how was it determined that the partition should be as stated? I have no doubt that it is correct, I simply don't understand it.

    Solution:

    Let \epsilon > 0 be given.
    Let P = {0, 1 - \epsilon/3, 1 + \epsilon/3, 2}

    Then L(f, P) = 0, since f(x) = 0 at points of each of these subintervals into which P subdivides [0, 2].

    Since f(1) = 1, we have:

    U(f, P) = 0(1 - \epsilon/3) + 1(2 \epsilon/3) + 0(2 - (1 + \epsilon/3))
    U(f, P) = (2 \epsilon)/3

    Hence, U(f, P) - L(f, P) < \epsilon and f is integrable on [0, 2]. Since L(f, P) = 0 for every partition,
    \int_0^2 \! f(x) \, dx = I_* = 0.
    The answer to that question is simple.
    Pick a partition that works. i.e makes the difference less than \epsilon
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2008
    Posts
    87
    Quote Originally Posted by Plato View Post
    The answer to that question is simple.
    Pick a partition that works. i.e makes the difference less than \epsilon
    So there are a lot of partitions that would work, then?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    May 2008
    Posts
    87
    Specifically, why is \epsilon divided by 3 chosen? Wouldn't also any number greater than 2 also yield that U(f, P) - L(f, P) < \epsilon?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,922
    Thanks
    1762
    Awards
    1
    Quote Originally Posted by posix_memalign View Post
    Specifically, why is \epsilon divided by 3 chosen? Wouldn't also any number greater than 2 also yield that U(f, P) - L(f, P) < \varepsilon?
    It could have been any \dfrac{\varepsilon }{N} where N>2.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Riemann Integral
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 8th 2012, 10:21 AM
  2. Sup & inf - riemann integral
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: June 5th 2010, 09:38 PM
  3. Riemann Sum to an integral
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 30th 2009, 11:33 AM
  4. Riemann Integral
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: August 17th 2009, 02:47 PM
  5. Replies: 6
    Last Post: August 4th 2007, 10:48 AM

Search Tags


/mathhelpforum @mathhelpforum