Evaluating Double Integrals: Switching the Order of Integration

  1. Alw

    Alw 8

    1. The problem statement, all variables and given/known data

    Evaluate the double integral by changing the order of integration in the iterated integral and evalutating the resulting iterated integral.

    2. Relevant equations

    [tex]\int[/tex][tex]^{1}_{0}[/tex] [tex]\int[/tex][tex]^{1}_{x}[/tex] cos(x/y)dydx

    3. The attempt at a solution

    I know how to solve a double integral after i've switched the order of integration, i'm having trouble with the acutal switching part :confused: The book we are using has one example in it regarding this, and it isn't very clear. If anyone would mind walking me through how to switch the order of integration, that'd be great :smile:

    Thanks in advance,
    -Andy

    edit: The text for the integrals didnt come out well, to make it more clear, its the integral from 0 - to - 1 and the integral
    from x - to - 1
     
  2. jcsd
  3. Gib Z

    Gib Z 3,348
    Homework Helper

    Why must you change the order? The integral is easily solvable as it is.
     
  4. siddharth

    siddharth 1,197
    Homework Helper
    Gold Member

    Is it? I can't see how.

    It always helps if you sketch the area over which you're integrating. Notice that the limits in x are from 0 to 1.

    So, the area over which you're integrating is bounded in the x direction by the lines x=0 and x=1. Also, since the limits in y are from x to 1, the boundaries in the y direction are the lines y=x and y=1. Can you sketch the area now? From this, can you figure out how to switch the order of integration?
     
  5. Gib Z

    Gib Z 3,348
    Homework Helper

    I read it wrong :( I seemed to read cos (y/x) >.<" Damn
     
  6. Alw

    Alw 8

    Ok, thanks! so if i'm not mistaken then, the new equation is:

    [tex]\int[/tex][tex]^{1}_{0}[/tex] [tex]\int[/tex] [tex]^{y}_{0}[/tex] cos(x/y)dxdy ?
     
  7. HallsofIvy

    HallsofIvy 40,297
    Staff Emeritus
    Science Advisor

    Yes. In your original integral x ranged from 0 to 1 and, for each x, y ranged from x to 1. That is the triangle with vertices (0,0), (1,1) and (0, 1). In the opposite order, to cover that triangle, y must range from 0 to 1 and, for each y, x must rage from 0 to y.
     
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