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Proof on a property of Riemann integrals

  1. Dec 3, 2008 #1
    1. The problem statement, all variables and given/known data

    Let f be continuous on [a,b] and suppose that [tex] f(x) \geq 0 [/tex] for all x Є [a,b]
    Prove that if there exists a point c Є [a,b] such that [tex] f(c) > 0 [/tex], then

    [tex] \int_{a}^{b} f > 0 [/tex]


    2. Relevant equations



    3. The attempt at a solution
    Using my books notation,

    Suppose P = {x0,x1,....,xn} is a partition of [a,b]. For each i = 1,...,n we let:
    [tex] M_i(f) = sup\{f(x) : x \epsilon [x_{i-1},x_i]\} [/tex]
    [tex] m_i(f) = inf\{f(x) : x \epsilon [x_{i-1},x_i]\} [/tex]

    The upper sum of f with respect to P: U(f,P)
    The lower sum of f with respect to P: L(f,P)

    The upper integral of f on [a,b]: U(f) = inf{ U(f,P) : P is a partition of [a,b] }
    The lower integral of f on [a,b]: L(f) = sup{ L(f,P) : P is a partition of [a,b] }

    Suppose there exists a c Є [a,b] such that f(c) > 0 then
    there exists a [tex] m_i(f) > 0 [/tex]
    thus L(f,P) > 0

    but then L(f) > 0, but since f is Riemann integrable:

    [tex] 0 < L(f) = U(f) = \int_{a}^{b} f [/tex]

    I know I'm kinda leaving out some details but is the outline alright? I have a feeling it's wrong since it was really short...
     
  2. jcsd
  3. Dec 3, 2008 #2
    Looks pretty decent to me.
     
  4. Dec 3, 2008 #3
    I should also include:

    [tex] \Delta x_i = x_i - x_{i-1} [/tex]

    [tex] U(f,P) = \sum_{i=1}^n M_i \Delta x_i [/tex]
    [tex] L(f,P) = \sum_{i=1}^n m_i \Delta x_i [/tex]
     
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