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Double integral problem

  • Thread starter cholyoake
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  • #1
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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.
 

Answers and Replies

  • #2
Curious3141
Homework Helper
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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.
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].
 
  • #3
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Oh of course, thank you very much for your help.
 
  • #4
DryRun
Gold Member
838
4
The question is:
[itex]\int_0^1\int_x^1e^\frac{x}{y}dydx[/itex]=[itex]\frac{1}{2}[/itex](e-1)
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?
 
  • #5
Curious3141
Homework Helper
2,843
87
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?
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.
 
  • #6
DryRun
Gold Member
838
4
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.
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.
 
  • #7
Curious3141
Homework Helper
2,843
87
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.
It's not a bug. It's just insufficient visual acuity to distinguish the cursive 'z' from the 'x' at that scale.
 
  • #8
DryRun
Gold Member
838
4
It's not a bug. It's just insufficient visual acuity to distinguish the cursive 'z' from the 'x' at that scale.
I guess there's no 'fixing' it then. Anyway, good to know it's not just me. :biggrin:
 
  • #9
HallsofIvy
Science Advisor
Homework Helper
41,833
955
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]
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]

=[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.
 
  • #10
Curious3141
Homework Helper
2,843
87
Oh of course, thank you very much for your help.
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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