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Homework Help: Resolution of a difficult Integral

  1. Apr 27, 2014 #1
    1. The problem statement, all variables and given/known data
    I need to solve this integral


    First limits are:


    Second limits are:


    3. The attempt at a solution


    I think it's getting too difficult to solve, is there an easier way to solve it?
    Last edited: Apr 27, 2014
  2. jcsd
  3. Apr 27, 2014 #2

    Simon Bridge

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    Why not try using substitutions?
  4. Apr 27, 2014 #3
    What do you mean with substitutions? Changing variables?
  5. Apr 27, 2014 #4

    Simon Bridge

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    Trig substitutions are popular.
    The idea is to get rid of that square-root sign when you go to evaluate the dz part.
  6. Apr 27, 2014 #5


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    cylindrical coordinates look helpful

    $$x\rightarrow r \, \cos(\theta)\\
    y\rightarrow r \, \sin(\theta)\\
    z\rightarrow z$$
  7. Apr 27, 2014 #6
    If I use cylindrical coordinates x=ρcos(Θ) ; y=ρsin(Θ) ; z=z

    I get the integral of ρ but how do I change the limits?
  8. Apr 27, 2014 #7

    Simon Bridge

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    i.e. if a<x<b, then the limits of ρ would be (a/cosθ)<ρ<(b/cosθ)
    ... and your next integral will be wrt θ

    A sketch of the cartesian limits will help you work out what the cylindrical limits should be.
  9. Apr 27, 2014 #8
    Excuse me, what is wrt?
  10. Apr 27, 2014 #9
    I get:




    And in the other:




    Is that correct?
  11. Apr 27, 2014 #10


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    wrt = 'with respect to'
  12. Apr 27, 2014 #11
    "with respect to"

    It's just a shorthand notation that we use. Similar to WLOG (without loss of generality) and iff (if and only if).
  13. Apr 27, 2014 #12

    Simon Bridge

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    ... and everybody leaps in to help with the easy question ;)

    - that would require me to do the problem, the idea is that you do it.
    I'll look in more detail a bit later - meantime:
    It's not always a blind substitution. Which order do you want to do the integration? Like this:
    $$\int \int \int \text{d}\theta \text{d}z \rho \text{d}\rho$$

    Note: if x^2 = z^2-y^2 then isn't y playing the role of a radius? Or is it z?
    Have you sketched out the region of integration?

    Are you unfamiliar with integrating in polar coordinates?

    Note: if this is an integral you constructed yourself rather than being handed to you in this form - you should reconsider the way you are dividing the volume up instead. In fact - that may be a good idea anyway.
    Last edited: Apr 27, 2014
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