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Double integral problem

  1. Aug 11, 2012 #1
    The question is:

    Show that:

    [itex]\int_0^1\int_x^1e^\frac{x}{y}dydx[/itex]=[itex]\frac{1}{2}[/itex](e-1)

    I've tried reversing the order of integration then solving from there:

    [itex]\int_0^1\int_y^1 e^{\frac{x}{y}}dxdy[/itex]

    =[itex]\int_0^1[ye^\frac{x}{y}]_y^1dy[/itex]

    =[itex]\int_0^1ye^\frac{1}{y}-ye^1dy[/itex]

    But I can't integrate [itex]ye^\frac{1}{y}[/itex]

    So either I've done something wrong when changing the order of integration or something else but I can't see how to go on from here.

    Thanks,
    Chris.
     
  2. jcsd
  3. Aug 11, 2012 #2

    Curious3141

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    Note that the integral can also equally well be represented by [itex]\int_0^1\int_0^xe^\frac{x}{y}dydx = \int_0^1\int_0^ye^\frac{x}{y}dxdy[/itex].
     
  4. Aug 11, 2012 #3
    Oh of course, thank you very much for your help.
     
  5. Aug 12, 2012 #4

    sharks

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    I have no idea why but my browser shows e^{z/y} instead of e^{x/y} as the itex code is written. I am using Firefox 14.0.1
    I tried clearing cache, etc but the problem still persists. Is this some bug with the forum?
     
  6. Aug 12, 2012 #5

    Curious3141

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    Try blowing up the font (Ctrl-'plus', using the Numeric Pad plus). It's an 'x', but looks like a 'z' at small font sizes.

    If you still see a 'z', then there's a real problem.
     
  7. Aug 12, 2012 #6

    sharks

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    That fixed it. But it is a really annoying problem. I don't want to leave my display font-size bigger than it should be in my browser in order to correctly read a problem/solution. This font-size bug will certainly cause a lot of doubts and confusion all around the forum.
     
  8. Aug 12, 2012 #7

    Curious3141

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    It's not a bug. It's just insufficient visual acuity to distinguish the cursive 'z' from the 'x' at that scale.
     
  9. Aug 12, 2012 #8

    sharks

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    I guess there's no 'fixing' it then. Anyway, good to know it's not just me. :biggrin:
     
  10. Aug 12, 2012 #9

    HallsofIvy

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    This is not a correct "reversal". In the original integral, x ranges from 0 to 1 and, for each x, y ranges from x up to 1. That is, it is the triangle above the line y= x. Reversing the order of integration, y goes from 0 to 1 and then, for each y, x goes from 0 to y. The integral is
    [tex]\int_0^1\int_0^y e^{\frac{x}{y}}dxdy[/tex]

     
  11. Aug 12, 2012 #10

    Curious3141

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    Please note that I just realised there is an error in my post. The first integral ([itex]\int_0^1\int_0^xe^\frac{x}{y}dydx[/itex]) should not be there as it defines the region below the line y = x (i.e. y ≤ x). So the only correct reversal is [itex]\int_0^1\int_0^ye^\frac{x}{y}dxdy[/itex], as HallsofIvy pointed out.
     
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