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Math Help - Algebra with Integral functions

  1. #1
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    Algebra with Integral functions

    I am trying to prove that if a<c<b, then \int _{a}^{b}f=\int _{a}^{c}f+\int _{c}^{b}f. This fact is very easy for me to visualize but I'm not sure how to get started.
    My professor talked about restricting the function and only evaluating the integral on a smaller interval. I feel like this might be key to proving it but I'm not sure how to tie it together. Any help would be appreciated.
    Last edited by zebra2147; February 9th 2011 at 08:26 AM. Reason: typo was noticed
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  2. #2
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    Quote Originally Posted by zebra2147 View Post
    I am trying to prove that if a<c<b, then \int _{a}^{b}f=\int _{a}^{c}f=\int _{c}^{b}f.
    The is a typo in your post. It must be \int _{a}^{b}f=\int _{a}^{c}f+\int _{c}^{b}f
    Consider a partition \mathcal{P} of [a,c] and a partition \mathcal{Q} of [c,b].
    Then \mathcal{P}\cup\mathcal{Q} is a partition of [a,b].

    On the other hand, if a partition \mathcal{R} of [a,b] does not contain c as a division point then simply add as a division point . Now you have two partitions, one of [a,c] and one of [c,b].

    That is an outline if an if and only if proof.
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