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Monotonicity of the riemann integral

  1. Apr 13, 2009 #1
    Hi everyone,

    For integrable [tex] f,g:\left[a,b\right]\rightarrow\mathbb{R} [/tex] with [tex]f(x)\leq g(x)[/tex] for all [tex]x\in\left[a,b\right][/tex], it's a basic property of the riemann integral that
    [tex]\[\int_a^b f(x)\,dx \leq \int_a^b g(x)\,dx\][/tex]

    My question is whether the strict version of this inequality holds, i.e. if we have the same hypotheses as above, except with [tex]f(x)<g(x)[/tex] for all [tex]x\in\left[a,b\right][/tex], then do we get the following inequality?
    [tex]\[\int_a^b f(x)\,dx < \int_a^b g(x)\,dx\][/tex]

    This question arose while trying to solve a rather different problem; I feel like it's not true in general, but I haven't yet come up with a counterexample.
     
  2. jcsd
  3. Apr 13, 2009 #2
    Yes, you get the strict inequality. The integral of a positive function is obviously positive.
     
  4. Apr 13, 2009 #3
    If you take a = b then the strict inequality is not true.
     
  5. Apr 14, 2009 #4
    Like jg89 pointed out, it holds as long as the lower and the upper limits of integration are not the same.
     
  6. Apr 15, 2009 #5
    Is this obvious?

    A Riemann-integrable function on [a, b] (with a < b) is continuous almost everywhere, so in particular it's continuous at one point; if this ensures that if the function is strictly positive, then its integral is positive.
     
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