1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Area under inexisting curve?

  1. May 18, 2014 #1
    Hi,
    I always wondered,
    what is the area under an inexisting curve.

    That arctan(1/sqroot(x^2-1)) for example. Its domain does not include from -1 to 1.

    If I take its integral from 0 to 10, what answer should I get?
     
  2. jcsd
  3. May 18, 2014 #2

    Mark44

    Staff: Mentor

    If the curve isn't defined, there's no region. Hence, area is meaningless.
    Whatever answer you get will almost certainly be wrong. For an integral to exist, the function in the integral must be continuous on the interval over which you're integrating. Your arctan function isn't defined on part of the interval [0, 10]. If you were to ignore these requirements for integration, whatever antiderivative you get won't be defined at 0.
     
  4. May 18, 2014 #3

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You could interpret it by allowing y to be complex. Whether that leads to a meaningful integral I'm not sure. I tried it for x2+y2=1, x from 1 to 2. I got terms like ##i \ln(2+\sqrt 3)##.
     
  5. May 20, 2014 #4
    Hmmm... you sure about that?
     
  6. May 21, 2014 #5

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Good catch. Even functions that would ordinarily be considered fairly pathological can be integrable. E.g. f(x) = 1 if x rational, 0 otherwise.
    Depends partly on what definition of integration is being used, as in Riemann-Stieltjes v. Lebesgue.
     
  7. May 21, 2014 #6
    No, I didn't think it was a particularly great catch. As I'm sure you know, there are extremely simple functions that have discontinuities and yet are integrable. A simple step function, for one.
     
  8. May 21, 2014 #7

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    I just excusing myself for not having read Mark44's post fully enough to have caught it:wink:.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Area under inexisting curve?
Loading...