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Homework Help: Improper Multiple Integrals

  1. Jan 27, 2008 #1
    I am having some trouble with the following 2 questions on improper multiple integrals. I hope that someone can help me out!

    1) Determine whether
    I=∫∫ cos(sqrt(x2+y2)) / (x2+y2) converges or diverges.
    x,y>1


    Solution:
    Let R=[0,1]x[0,1]
    B(0,1)=ball of radius 1 centered at origin
    R+xR+=positive xy-plane
    f(x,y)=the integrand

    The solution says that
    I=∫∫ f(x,y) dA - ∫∫ f(x,y) dA
    R+xR+ \B(0,1) R\B(0,1)

    And then showed that both integrals converge, so the given improper integral I converges.
    ============================
    Now, I am having a lot of trouble understanding the red part, WHY is it true?



    ============================
    2) Determine whether the following converges or diverges.
    Let S=[-1,1]x[-1,1]
    2a)
    ∫∫ x2 / (x2+y2) dA
    S
    2b)
    ∫∫ sqrt|x| / (x2+y2) dA
    S

    ============================
    In the solutions manual, seemingly, they know the answer at the beginning; they inscribed a circle within the rectangle S for 2a) and inscribed the rectangle S in a circle for 2b), said that the integrand >0 except the origin, and used the comparsion test to conclude the first one diverges and the second converges.
    ============================
    Now, I just want to know HOW I can get a first feeling about whether the above improper integrals will converge or diverge before going into the details. It's nice to know the answer ahead of time, so that I can know which direction to push forward the proof. Otherwise, I will just be doubling my amount of time and effort to finish.

    Thank you for explaining!
     
    Last edited: Jan 27, 2008
  2. jcsd
  3. Jan 27, 2008 #2
    I'm not sure exactly what the red part for 1) is saying. Try converting it to cylindrical coordinates and set [tex]R = \{\theta \in [0,2\pi], r \in (0,1] \}[/tex]. You'll basically end up needing to show that

    [tex]\int^1_0 \frac{\cos r}{r}dr[/tex]

    converges.
     
  4. Jan 28, 2008 #3
    The red part:
    I=∫∫ f(x,y) dA - ∫∫ f(x,y) dA
    R+xR+ \B(0,1) R\B(0,1)

    R+xR+ \B(0,1), (xy-plane take away ball centered at origin of radius 1), is region of integration of the first double integral

    R\B(0,1), [0,1] x [0,1] take away ball centered at origin of radius 1, is region of integration of the second double integral

    But I don't understand the equality in the red part.




    About your method:

    In cylindrical coordinates, the limits of integration are not going to be "nice" since the region is x>1 and y>1. How can we describe this in cylindrical coordinates?? Besides, it's a double integral, how can cylindrical coordinates work in R^2?
     
    Last edited: Jan 29, 2008
  5. Jan 28, 2008 #4

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    r goes from 1 to infinity so it's then just a single variable convergence problem.
     
  6. Jan 29, 2008 #5
    I don't think r goes from 1 to infinity since the region x,y>1 is not a ring...

    Can someone please at least help me with one of the 2 questions? Any help of any kind would be appreciated!
     
  7. Jan 29, 2008 #6
    1) I think the solutions have mistaken the interpretation of x,y>1, they think that x,y>1 is the region in the positive xy-plane take away [0,1]x[0,1], but this is clearly not the case, so the solution is wrong.

    How can we actually solve this problem?
     
  8. Jan 29, 2008 #7
    Last edited by a moderator: Apr 23, 2017
  9. Jan 30, 2008 #8
    1) But in our case, since it's a double integral (2-D), it's impossible to use a 3-D cylindrical coordinates

    Also, how can we possibly know whether the integral converges or diverges? If we don't know what to prove, how can we solve it?
     
    Last edited by a moderator: Apr 23, 2017
  10. Jan 30, 2008 #9
    2) Any help with this problem? I have the answers to these, but I don't understand how they figured out convergence/divergence
     
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