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Area of region

  1. Jan 20, 2014 #1

    etf

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    Here is my task and attempt for solution:

    attempt.gif

    As you can see, I tried with substitutions but I have no idea what to do next.
    Any suggestion?
     
    Last edited: Jan 20, 2014
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  3. Jan 21, 2014 #2

    haruspex

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    The equation you start with looks nasty. Are you sure it's not supposed to be x3+8y3=x2+4y2, or somesuch?
    Either way, I can't see the merit of that substitution. Why not just go to the usual polar coordinates? At least you can get as far as writing the area as a single integral. Still tough though.
     
  4. Jan 21, 2014 #3

    etf

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    I'm sure it's how I wrote it... I tried usual polar coordinates too and I got that r=(4*cos(phi)^2+sin(phi)^2)/(cos(phi)^3+8*sin(phi)^3). I don't think that this substitution is more useful than previous. I tried with few other substitutions in order to make region easier but without success. I really have no idea what to do here :(
     
    Last edited: Jan 21, 2014
  5. Jan 21, 2014 #4

    haruspex

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    I agree it's still nasty, but at least you can write out the integral for the area now . With your other substitution you'd have to calculate quite a messy Jacobian to get that far. (Did you compute the Jacobian? Maybe it simplifies magically, but I doubt it.)
     
  6. Jan 22, 2014 #5

    haruspex

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    I've had maybe a better idea. Try u = x, v = y/x.
     
  7. Jan 23, 2014 #6

    etf

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    Wow! This one is key to my problem :)
    If I didn't make mistake somewhere, area should be:

    povrsina1.gif
     
  8. Jan 23, 2014 #7

    haruspex

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    Looks good until you get to the last line. A = ∫∫dxdy = ∫∫Jdudv etc. I'm afraid it's going to get quite complicated, but it's all doable.
     
  9. Jan 23, 2014 #8

    etf

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    I computed it using Matlab, I didn't try to do it by hand...
    Thanks a lot!
     
  10. Jan 23, 2014 #9

    haruspex

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    OK, I see - I misread your double integral as two separate integrals. (That's not the way I write them.)
    The simplicity of the answer does suggest we've missed a trick somewhere.
     
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