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Math Help - continuous function, integral

  1. #1
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    continuous function, integral

    Suppose that f: [a,b] \rightarrow \mathbb{R} is continuous and f(x) \geq 0 for all  x \in [a,b]. Prove that if \int^b_a f(x)dx=0, then f(x)=0 for all  x \in [a,b].
    Attempt
    I had attempted to do this problem by contradiction, except I did not understand how to finish the problem. I would appreciate a few helpful hints on this one.
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  2. #2
    Newbie aleph1's Avatar
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    Fundamental Proof

    Quote Originally Posted by selenne431 View Post
    Suppose that f: [a,b] \rightarrow \mathbb{R} is continuous and f(x) \geq 0 for all  x \in [a,b]. Prove that if \int^b_a f(x)dx=0, then f(x)=0 for all  x \in [a,b].
    From the proof in the fundamental theorem of calculus:

    F(b) - F(a) = \lim_{\| \Delta \| \to 0} \sum_{i=1}^n \,[f(c_i)(\Delta x_i)]\

    F(b) - F(a) = \int_{a}^{b} f(x)\,dx\

    With f(c_i)\geq 0 from given f(x) \geq 0 for all  x \in [a,b]

    And the definite integral evaluated as \int^b_a f(x)dx=0 from given, then
    \lim_{\| \Delta \| \to 0} \sum_{i=1}^n \,[f(c_i)(\Delta x_i)]=0\

    with f(c_i) confined to be non negative, the summation can only be zero if f(x)=0 for all  x \in [a,b].
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