1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Use Upper And Lower Sums To Evaluate An Integral

  1. Aug 27, 2010 #1
    1. The problem statement, all variables and given/known data

    The question is to use upper and lowers sums, Un and Ln, on a regular petition of the intervals to find the integral from 1 to 2 of f(x) = [[x]], where [[x]] is the greatest integer function.


    2. Relevant equations

    [tex]\Delta[/tex]x = [tex]\frac{b-a}{n}[/tex]

    3. The attempt at a solution

    [tex]\Delta[/tex]x = [tex]\frac{2-1}{n}[/tex] = [tex]\frac{1}{n}[/tex]

    The minimum and maximum of f(x) on every subinterval of [1,2] would be 1 except for the subinterval which includes x=2 where the maximum value of f(x)=2, so for f(x) there exists a unique number I such that Ln [tex]\leq[/tex] I [tex]\leq[/tex] Un

    I know that the area will be 1, as for [1,2) Un = Ln = [tex]\Delta[/tex]x[tex]\sum[/tex]1 = [tex]\frac{1}{n}[/tex] x 1n = 1

    However I am not sure how to include the subinterval that contains x=2 into my calculations. Any help on this would be great!

    Thanks!
     
  2. jcsd
  3. Aug 27, 2010 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    If you have n subintervals, evenly spaced say, then n-1 will have f(x) be just 1, and one will contain a point where f(x) is 2. So your lower sum, using the lowest value on each interval, will just give you the area of 1 as expected. For the upper sum, you have

    [tex](\sum_{i=1}^{n-1} \frac{1}{n})+\frac{2}{n}[/tex] adding up all the parts where the upper bound is 1, plus the one part where the upper bound is 2. Can you find the limit of that summation as n goes to infinity? (Of course we know it should be 1)
     
  4. Aug 27, 2010 #3
    That makes perfect sense! Thanks a bunch :D
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook