How Do I Solve a Complex Double Integral Problem?

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In summary, the equation you start with looks nasty, but you can write out the area as a single integral with the substitution u=x,v=y/x.
  • #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?
 
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  • #2
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.
 
  • #3
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 :(
 
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  • #4
etf said:
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 :(
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.)
 
  • #5
I've had maybe a better idea. Try u = x, v = y/x.
 
  • #6
Wow! This one is key to my problem :)
If I didn't make mistake somewhere, area should be:

povrsina1.gif
 
  • #7
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.
 
  • #8
I computed it using Matlab, I didn't try to do it by hand...
Thanks a lot!
 
  • #9
etf said:
I computed it using Matlab, I didn't try to do it by hand...
Thanks a lot!
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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