1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Definite Integral

  1. Dec 10, 2004 #1

    I need help doing the following integration:


    where n is an integer and [.] denotes the greatest integer function (floor), i.e. [x] = greatest integer less than or equal to x.

    The answer given in the book is


    whereas I am getting zero as the answer. My solution is a bit jerky due to the step marked # below:

    By definition, there exists some [tex]k \epsilon Z[/tex] such that

    [tex]k\leq x < k+1[/tex] (so that [x] = k)

    which means that

    [tex]k-1<k-\frac{1}{\sqrt{2}}\leq x - \frac{1}{\sqrt{2}} < k+1-\frac{1}{\sqrt{2}}[/tex] and
    [tex]k-1<k-\frac{1}{\sqrt{3}}\leq x - \frac{1}{\sqrt{3}} < k+1-\frac{1}{\sqrt{3}}[/tex]

    But this would mean that (#)

    [tex][{x-\frac{1}{\sqrt{2}}] = k-1[/tex]
    [tex][{x-\frac{1}{\sqrt{3}}] = k-1[/tex]

    Making the integrand zero and hence the integral zero as well.

    I am not sure if this reasoning is correct (in particular, the integer parts cannot be greater than k-1 so this step could be wrong but still they can attain no other integral value) so I would be very grateful if someone could guide me here.

    Thanks and cheers
    Last edited: Dec 10, 2004
  2. jcsd
  3. Dec 10, 2004 #2


    User Avatar
    Science Advisor
    Homework Helper

    I suggest another method of approach. Draw the graph. Once you have that, the answer is trivial.

    Notice that
    [tex]\lfloor x-\sqrt2 \rfloor=\lfloor x-\sqrt3 \rfloor[/tex] if [tex]0\leq x < \sqrt{2}[/tex] or [tex]\sqrt{3}\leq x \leq 1[/tex]

    What happens when [tex]\sqrt{2} \leq x < \sqrt{3}[/tex].

    What changes when x increases by 1?
    Last edited: Dec 10, 2004
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Definite Integral
  1. Definite integral (Replies: 1)

  2. Definite integral (Replies: 7)