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Homework Help: Double integral confused

  1. Aug 2, 2010 #1
    i am confused about the double integral ʃʃ cos(x+2y)dA, where R = [0,pi]x[0,pi/2]
    i realize for the integral that i must do u-substitution. when i do this, however, do i also have to change the boundary conditions as in a single integral?

    i got -8 without changing the boundary conditions, but i'm not sure if that's right.
    i will show my work if anyone wants to check how i got there
     
  2. jcsd
  3. Aug 2, 2010 #2

    Dick

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    The u-substitutions you have to do are pretty easy. But I don't get -8. Maybe you'd better show how you did it.
     
  4. Aug 2, 2010 #3
    I did this out quickly using [tex] \cos u ={{{\rm e}^{{\rm j} u} + {\rm e}^{-{\rm j} u}}\over{2}}[/tex], and didn't get -8 either. It's probably just a simple error in the substitution process.
     
  5. Aug 2, 2010 #4
    Why not just use the sum formula for cosine?

    [tex] cos(A + B) = cosAcosB - sinAsinB. [/tex]
     
  6. Aug 2, 2010 #5
    Just used the first thought that came to my mind. It is so simple by either method, and I didn't bother to consider all methods to find the best. I just wanted to get a number to help the OP know if he was right or wrong. Actually, Dick beat me too it, so I just provided a second verification.
     
  7. Aug 2, 2010 #6
    hm, it seems no one answered my question: do i have to change the boundary conditions when doing the u-substitution?
     
  8. Aug 2, 2010 #7
    yes, you do. out of curiosity, which u substitution did you use?
     
  9. Aug 2, 2010 #8

    Dick

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    Integrate dx first and dy second, if that's your choice of order. You'll need a u-substitution for each, and yes, you may need a change of limits. Depending on how you do it. It's kind of useless to debate this until you show how you arrived at the wrong answer.
     
  10. Aug 2, 2010 #9
    using the sum formula for cosine i got -2?
     
    Last edited: Aug 2, 2010
  11. Aug 2, 2010 #10
    That looks correct
     
  12. Aug 2, 2010 #11
    on a somewhat unrelated note, is ln2 the correct answer for ʃʃ (xe^x)/y dydx for R = [0,1]X[1,2]? i did integration by parts for the xe^x part but had the same issue of whether or not to change the start/end point.
     
  13. Aug 2, 2010 #12
    Looks correct to me.
     
  14. Aug 2, 2010 #13
    Yes, this seems to be a question in your mind. The simple rule is that you only need to change limits if you have a substitution that results in a change of variables. If your functions are still using x and y, and your integration is still over dx and dy, then there is no need to think about changing limits. However, if you change variables, such as u=2x, and/or w=sin(y), then you will generally need to change the limits.
     
  15. Aug 2, 2010 #14
    thank you
     
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