wu :: forums « wu :: forums - Odd integration problem » Welcome, Guest. Please Login or Register. Jun 9th, 2023, 7:00am RIDDLES SITE WRITE MATH! Home Help Search Members Login Register wu :: forums  riddles  putnam exam (pure math) (Moderators: SMQ, Icarus, Grimbal, william wu, Eigenray, towr)  Odd integration problem « Previous topic | Next topic » Author Topic: Odd integration problem  (Read 807 times)
NightBreeze
Newbie   Posts: 26 Odd integration problem   « on: Jun 21st, 2008, 11:17am » Quote Modify

Evaluate      {x/y}{y/x} dxdy

where {x} denotes the fractional part of x: {x} = x - x « Last Edit: Jun 25th, 2008, 3:47pm by NightBreeze » IP Logged
Miles
Junior Member   Cambridge, England

Gender: Posts: 95 Re: Odd integation problem   « Reply #1 on: Jun 24th, 2008, 7:00am » Quote Modify

 hidden: The trick is to find a convenient way to cut up the unit square (0 < x < 1, 0 < y < 1) and sum the integral on each piece.   By symmetry, we can integrate over the half of the square where x > y and double the result.  This ensures that {y/x} = y/x.  We chop this half into regions bounded by y = x / n & y = x / (n+1) & x = 1, for each integer n >=1.  Note that in this region, n < x/y < n+1 so {x/y} = x/y - n.  Putting it all together the result we want is          the sum for n>=1          of the integral of        2.(y/x).(x/y - n) = 2(1 - ny/x)        with respect to x and y        over the region where 0 < x < 1, x/n < y < x/(n+1).   The integration gives me 1/2n - 1/(n+1) + n / (n+1)^2 which rearranges to      (1/2).[1/n - 1/(n+1) - 1/(n+1)^2]   On summing over n>=1, the first two terms in the sum telescope into 1/2 and using a standard result the last term gives   -(1/2).(pi^2 / 6 - 1)

So the answer is 1 - pi^2 / 12. IP Logged
NightBreeze
Newbie   Posts: 26 Re: Odd integation problem   « Reply #2 on: Jun 24th, 2008, 10:03am » Quote Modify

Correct. IP Logged

 Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »