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Calculate the double integral : int int xye^((x^2)(y))

  1. Feb 7, 2013 #1
    1. The problem statement, all variables and given/known data

    Calculate the double integral

    int int xye^((x^2)(y)) , 0<= x <= 1 , 0<= y <= 2

    2. Relevant equations

    Integral by parts

    uv - int vdu

    3. The attempt at a solution

    IMG_20130207_073836.jpg

    The answer in the back of the book is (1/2)((e^2) -3) , but I get (1/2)((e^2) -1) .

    I think I made a positive/negative sign error, but I can't find it. I've had similar encounters where I would get close to the answer for questions involving integration by parts. I might of made a consistent error in one of my lines for each other question involving integration by parts, but I don't know where.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 7, 2013 #2
    Your error is in the first step
     
  4. Feb 7, 2013 #3
    You have a ##dx## on the first line. That means you integrate with respect to x and hold y constant.

    ##\int\limits_{0}^1 xye^{x^2y} \ dx = y\int\limits_{0}^1 xe^{x^2y}\ dx##

    Does this make it clearer?
     
  5. Feb 7, 2013 #4

    Not really, I already knew that I should think of y as a constant when integrating with respect to x, but I rarely ever factor it out and just imagine that y is a number, an integrate the function.

    But I think I know where I made my mistake:
    The derivative of e^((x^2)(y)) = (2xy)(e^((x^2)(y))) , and I forgot to take out the y in my answer.

    Thanks!
     
  6. Feb 7, 2013 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    That is, by the way, the hard way to do this problem. Change the order of integration:
    [tex]\int_{y=0}^2\int_{x=0}^1 xye^{x^2y}dx dy[/tex]
    Let [itex]u=x^2y[/itex] so that [itex]du= 2xydy[/itex] and so [itex](1/2)du= xydy[/itex]. That simplifies the problem a lot!
     
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