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!

Integrating mod function

  1. Dec 3, 2016 #1
    1. The problem statement, all variables and given/known data
    I need to integrate
    ##\langle |v_x| \rangle = \int^{\infty}_0 |v_x| \sqrt{\frac{m}{2\pi kT}}e^{-mv_x^2/2kT}dv##
    For context this is a Maxwell boltzmann distribution in one dimension, and I've actually been asked to calculate ##\langle v_x \rangle## which is given by ##|v_x|f(v)## where ##f(v) = \sqrt{\frac{m}{2\pi kT}}e^{-mv_x^2/2kT}## is the Maxwell Boltzmann distribution in the x-direction. Not sure if the question is best put in physics or maths.
    2. Relevant equations


    3. The attempt at a solution
    I'm a little confused because since ##v_x## is always positive between infinity and zero (I think?) the integral is actually just ##\langle v_x \rangle##, since the mod can be ignored if it's always positive. That can't be it though, I've already been asked to calculate ##\langle v_x \rangle## in the first part of the same question. So I suppose my treatment of the mod must be wrong.

    Thanks for any help!
     
    Last edited: Dec 3, 2016
  2. jcsd
  3. Dec 3, 2016 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    What was your result for ##\langle v_x \rangle##?

    Do you see why ##\langle v_x \rangle \neq \langle |v_x| \rangle##?

    What slight (but important) error have you made in your expression for ## \langle |v_x| \rangle##?
     
  4. Dec 3, 2016 #3
    Is the error related to the limits, by any chance? Should probably go to ##-\infty## and that would explain a lot.
     
    Last edited: Dec 3, 2016
  5. Dec 3, 2016 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Try it, to see what you get.

    Anyway, what is your answer to my question about ##\langle v_x \rangle##?
     
  6. Dec 3, 2016 #5
    It is the limits, velocity can be negative, I'd been working with speed in 3D so only needed to integrate ##\infty## to zero. Didn't read the question carefully so missed that we had moved to velocity distribution. As to ##\langle v_x \rangle##, I have that equal to zero.
     
  7. Dec 3, 2016 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Right: ##\langle v_x \rangle = 0##, and this follows immediately (with essentially no work) from the fact that ##f(v_x)## is an even function of ##v_x##, so ##v_x f(v_x)## is an odd function. In contrast, ##|v_x| f(v_x)## is an even function of ##v_x##, so its integral over the whole line can be easily related to its integral over the half-line ##\{ v_x \geq 0 \}##.
     
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: Integrating mod function
  1. Integrate a function (Replies: 12)

Loading...