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Homework Help: Double integral (need help and checking)

  1. Feb 21, 2010 #1
    1. The problem statement, all variables and given/known data
    Let Ω ⊂ R^2 be the parallelogram with vertices at (1,0), (3,-1), (4,0) and (2,1). Evaluate ∫∫_Ω e^x dxdy.

    Hint: It may be helpful to transform the integral by a suitable (affine) linear change of variables.


    2. Relevant equations



    3. The attempt at a solution
    Ok here is what I have done:
    From the sketch of the parallelogram, I have found the limits to be x=y+1 to (4-2y) and y=-1 to 1. With this, I am able to determine the solution of the integral which is 3/2*e^2 -1 -1/2*e^6.

    Could anyone please verify this for me? Also, if my solution turns out to be right, how would I approach the hint to find the solution of this double integral? Thanks.
     
  2. jcsd
  3. Feb 21, 2010 #2
    From what I understand, you need to separate the parallelogram into two areas when you calculate this double integral.
    What you did was that you took only the upper part of the parallelogram .

    you have 4 different sides which you have to take into account.

    I think this is how you need to do it...
     
  4. Feb 21, 2010 #3
    Right, so should my integral look like as follows?

    ∫∫ e^x dxdy with x=4-2y to y+1, y=0 to 1 + ∫∫ e^x dxdy with x=1-2y to y+4, y=-1 to 0
     
  5. Feb 21, 2010 #4
    I think it should be this:

    ∫∫ e^x dxdy with x=y+1 to 4-2y, y=0 to 1 + ∫∫ e^x dxdy with x=1-2y to 3y+4, y=-1 to 0
     
  6. Feb 21, 2010 #5
    You are right for the limits of x but I think the y limits should be the way I first wrote it? Did you calculate the gradient wrongly? Btw, thanks for your help so far.
     
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