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Math Help - Average Value of an abs. value function

  1. #1
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    Average Value of an abs. value function

    Q: Find the average value of  f(x) = | 7 - x |    [4,10]

    My solution: I used  \frac {1}{b-a} \int_{a}^{b} f(x)dx

    So I have  \frac {1}{10-4} \int_{4}^{10} (7 - x) dx , but the result is 0...

    KK
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  2. #2
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    You need to find, first
    \int_4^{10}|7-x|dx
    The function is continous therefore you can subdivide this in any way you wish,
    Do this,
    \int_4^7 |7-x|dx+\int_7^{10}|7-x|dx
    Now on the interval,
    f(x)=|7-x|=7-x on [4,7]
    And,
    f(x)=|7-x|=x-7 on [7,10]
    Thus,
    \int_4^7 7-x dx+\int_7^{10} x-7dx
    --------
    the result is 0...
    Impossible, since the function is continous and,
    |7-x|>0
    Then the dominance rule says,
    \int_4^{10}|7-x|dx>0
    And when you divide a non-zero number by the length of this interval you do not end with a negative number
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