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Change of variable integral?

  1. Jan 8, 2016 #1
    1. The problem statement, all variables and given/known data
    Evaluate the integral

    ∫∫sin(x+y)/(x+y) dydx over the region D

    whereD⊆R2 is bounded by x+y=1, x+y=2, x-axis, and y-axis.


    2. Relevant equations


    3. The attempt at a solution
    I think that I need to use a change of variables but can not find any change of variables that work. One thing I thought would work is using u = sin(x+y) and v = (x+y) but finding the jacobian doesn't work with the u transformation.
    This leads me to think I might need to do a taylor expansion instead.

    Any help with a recommended change of variables or a step in the right way would be appreciated.
     
  2. jcsd
  3. Jan 9, 2016 #2

    haruspex

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    Try something much simpler, but along similar lines.
     
  4. Jan 9, 2016 #3

    vela

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    Did you make a sketch of D? If not, do so. Add to it lines of constant v=x+y to see why this is a good choice. That might give you an idea of what to use for u.
     
  5. Jan 9, 2016 #4
    I sketched D and put in the constant v=x+y but still can not see any good change of variable for u. The sin is really causing me problems. I constantly think I need to make my "u' have the sin term in it but can not come up with anything that works
     
  6. Jan 9, 2016 #5

    haruspex

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    Given the substitution v=x+y, what is the most obvious way to define u? (It does not involve a trig function.)
     
  7. Jan 9, 2016 #6

    vela

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    You don't want the sine in the transformation. That's what haruspex was implying when he said to try something simpler.
     
  8. Jan 9, 2016 #7
    1. Looking at my graph is seems to be convenient if I define u to be a constant, possibly u = 2? ... When I learnt change of variables I was told I would never need to come up with the change of variable myself so that is why I'm having some troubles. I appreciate the help.
     
  9. Jan 9, 2016 #8

    haruspex

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    What expression for u would result in the u, v coordinate system being Cartesian?
     
  10. Jan 10, 2016 #9
    I still am completely stuck. Any way I look at it I get stuck with something I can't integrate with the sin in the integral. I am working off the assumption that v = x+y is correct. This would make my integral something like ∫ sin(v)/v . My last thought is to make u = 1/x+y. But this still seems to give me some integral that I can't solve...
     
  11. Jan 10, 2016 #10

    haruspex

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    Isn't u=y-x the most obvious choice, by a mile?
    If you think that doesn't help, integrate wrt u first, making sure you get the limits right. They depend on v.
     
  12. Jan 10, 2016 #11
    That change of variable makes sense when I look on it on a graph, thanks. Now for my bounds I get, 1≤v≤2 and -v≤u≤v . I worked out the integral to just become sin(v). Thanks for the help!
     
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