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Math Help - Integral

  1. #1
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    Exclamation Integral

    Let f be integrable in [a,b]. Prove that:

     <br />
\int_{a}^{b} |f| \leq{\displaystyle\int_{a}^{b}|f|}<br />

    Thanks
    Last edited by osodud; December 10th 2009 at 08:19 PM.
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by osodud View Post
    Let f be integrable in [a,b]. Prove that:

     <br />
\int_{a}^{b} f| \leq{\displaystyle\int_{a}^{b}|f|}<br />

    Thanks
    I don't understand what you mean.
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  3. #3
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    Quote Originally Posted by VonNemo19 View Post
    I don't understand what you mean.
    I have edited the expression.

    Check it please
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by osodud View Post
    Let f be integrable in [a,b]. Prove that:

     <br />
\int_{a}^{b} |f| \leq{\displaystyle\int_{a}^{b}|f|}<br />

    Thanks
    Something doesn't seem right here...do you possibly mean

    \int_{a}^{b} |f| \leq\left|\int_{a}^{b}f\right|

    or

    \left|\int_{a}^{b} f\right| \leq\int_{a}^{b}\left|f\right|
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  5. #5
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    Quote Originally Posted by Chris L T521 View Post
    Something doesn't seem right here...do you possibly mean

    \left|\int_{a}^{b} f\right| \leq\int_{a}^{b}\left|f\right|
    Yesˇˇ That one
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    \left|\int_{a}^{b} f\right| \leq\int_{a}^{b}\left|f\right|
    Quote Originally Posted by osodud View Post
    Yesˇˇ That one
    First note that by the FTC, \int_a^b f= F(b)-F(a).

    Thus, \left|\int_a^b f\right|=\left|F(b)-F(a)\right|.

    Applying the triangle inequality, we have

    \begin{aligned}\left|F(b)-F(a)\right| &\leq \left|F(b)\right|+\left|F(a)\right|\\ &\leq F(a)+F(b)\\ &\leq F(a)-F(x_1)+F(x_2)-F(x_1)+\ldots-F(x_{k-1})+F(b)\\ &=-(F(x_1)-F(a))+(F(x_2)-F(x_1))+\ldots+(F(b)-F(x_{k-1}))\\ &=-\int_a^{x_1}f+\int_{x_1}^{x_2}f-\int_{x_2}^{x_3}+\ldots+\int_{x_{k-1}}^bf\\ &=\int_a^b\left|f\right|\end{aligned}

    I think that's the way to approach it, but I'll let others chip in (or inform me I'm not correct)
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