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Math Help - Double integral over a triangular domain; getting different answers for x or y simple

  1. #1
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    Double integral over a triangular domain; getting different answers for x or y simple

    Hello all. Been a lurker of the forums for a little while, thought it was time to finally join up in hopes that I can both give and receive some guidance.

    The particular problem i've been stuck on four hours is as follows (it was just a practice problem, it isn't for any assignment or anything). I'm not really sure how to use the actual math notation on here, if somebody could point me towards a help thread for that or something that would be great if there are any obvious errors in the notation.

    Q: Evaluate the volume between the graph of f(x,y)=exp(x-y) and the domain D, where D is the interior of the triangle with vertices (0,0), (1,3) and (2,2).
    \int\int_{D}exp(x-y))dxdy, where D is described above.

    I first tried the problem by calling the domain y-simple. From a quick sketch I got my bounds of integration (phi1(x) and phi2(x))as y=x and y=3x, so the iterated integral was

    \int_{0}^{2}\int_{x}^{3x}exp(x-y)dydx, which I solve to be (1/2)[3+exp(-4)]

    I then tried the problem treating the domain as x-simple, which gave me the following iterated integral:
    \int_{0}^{2}\int_{y/3}^{y}exp(x-y)dxdy, which I solve to be (3/2)[1+exp(-2)]. Wolphram Alpha has given me the same solutions to the respective integrals. I am not sure how I can illustrate the way in which I obtained those, I'm very new to this LaTex stuff. I drew diagrams of that triangle in the xy plane, and then determined the simple equations of the lines from (0,0) to the respective vertices, as those are the functions that bound x or y on our domain. I hope I explained that clearly enough.


    The volume of the solid should be the same no matter which way I treat the domain, so I must be making an error in the way I set up the iterated integrals. I've been getting the earlier problems in the textbook right, and am unable to find my mistake. Could anyone point me in the right direction or tell me where to look for my mistake without just giving a free answer?
    Thank you very much.
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  2. #2
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    Re: Double integral over a triangular domain; getting different answers for x or y si

    You should first attempt to get the correct area.

    Why are you trying to do it in one piece?

    \int_{0}^{1}\int_{x}^{3x}e^{x-y}\;dy\;dx

    The other piece is:

    \int_{1}^{2}\int_{x}^{4-x}e^{x-y}\;dy\;dx

    Reversing the order of integration will not help. You'll still need two pieces.

    If you REALLY wanted to do it in Polar Coordinates, THEN you could take it out in one piece.
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  3. #3
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    Re: Double integral over a triangular domain; getting different answers for x or y si

    It is also useful to note that e^{x- y}= e^xe^{-y}.
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