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!

Homework Help: Triangle Inequality for integrals proof

  1. Oct 16, 2011 #1
    1. The problem statement, all variables and given/known data
    What I want to show is this:
    ∫|x+y| ≤ ∫|x| + ∫|y|

    2. Relevant equations
    |x+y| ≤ |x| + |y|

    3. The attempt at a solution

    So I thought if I used the triangle inequality I could get to something along the lines of:

    Lets g belong to the real numbers
    ∫|x+y| = ∫|x+g-g+y|≤ ∫|x+g| + |y-g|= ∫|x+g| + ∫|y-g|

    As g belongs to the reals it can be zero meaning ∫|x+y| ≤ ∫|x| + ∫|y|.

    Now the problem with this is that is uses the triangle inequality and I have no idea if the triangle inequality works this way, and if it does I need to prove it, and I have no idea about where to start that from.
  2. jcsd
  3. May 13, 2013 #2


    User Avatar
    Gold Member

    I suppose I am necro-posting here, but this result follows from two facts. One is the linearity of integration:
    [tex]\int (f(x)+g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx,[/tex]
    and the other is that integration preserves inequalities: if [itex]f(x) \le g(x)[/itex] on the interval [itex] [a,b] [/itex], then
    [tex] \int_{a}^{b}f(x) \, dx \le \int_{a}^{b} g(x) \, dx. [/tex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted