Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

An integral paradox ?

  1. Jun 9, 2010 #1
    An integral paradox ??

    let be [tex] \int_{0}^{\infty}xdx \int_{0}^{\infty}ydy [/tex]

    changing to polar coordinates we get that the double integral above shoudl be

    [tex] 2\int_{0}^{\infty}r^{3}dr [/tex]

    althoguh they are all divergent , is this true can we ALWAYS make a change of variable to polar coordinates without any ambiguity ??
     
  2. jcsd
  3. Jun 9, 2010 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Re: An integral paradox ??

    One can always change from rectangular to polar coordinates. However the transformation you gave is incorrect. The coefficient is not 2 but 1/2.
     
  4. Jun 10, 2010 #3
    Re: An integral paradox ??

    am.. thanks a lot

    but my question is, the Area of a Circle is NOT equal to the area of an Square [tex] \frac{C}{S}= \pi [/tex]

    hence , how could we be completely sure [tex] \iint _{C} f(x,y)dxdy = \iint _{S} f(x,y)dxdy [/tex]
     
  5. Jun 10, 2010 #4

    Gib Z

    User Avatar
    Homework Helper

    Re: An integral paradox ??

    Rectangular coordinates don't necessarily trace out rectangles and Polar coordinates don't necessarily trace out Circles in the xy plane. The path they trace out is predetermined by a rule, eg To describe the path of the unit circle in rectangular coordinates we say x^2+y^2 = 1, and the same path could be described in polar coordinates with x= cos t, y= sin t, t varies from 0 to 2pi.

    It's your job to change the bounds and integrand of the integral accordingly when change coordinates so that they still sum the same overall function values over the same domain.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: An integral paradox ?
  1. A paradox? (Replies: 11)

  2. On Integration (Replies: 4)

  3. An integral (Replies: 2)

Loading...