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Math Help - Integral Proof

  1. #1
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    Integral Proof

    I have been asked to justify if the following statement is true or false.

    'If the integral of f(x) >= 0 on [a,b], then f(x) is >= 0 on [a,b].

    Would (8-x^2) over the domain [-4,4] be a counter example to this? thus justifying that the statement is false.

    I beleive so as f(x) is <= 0 for some values but the area over that domain is positive. Can anyone tell me if iam on the right path?
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  2. #2
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    Quote Originally Posted by olski1 View Post
    I have been asked to justify if the following statement is true or false.

    'If the integral of f(x) >= 0 on [a,b], then f(x) is >= 0 on [a,b].

    Would (8-x^2) over the domain [-4,4] be a counter example to this? thus justifying that the statement is false.

    I beleive so as f(x) is <= 0 for some values but the area over that domain is positive. Can anyone tell me if iam on the right path?
    8 - x^2 is NOT \geq 0 for the entire domain [-4, 4].
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    So doesnt that proove that the statement is false?
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    Quote Originally Posted by olski1 View Post
    So doesnt that proove that the statement is false?
    Definitely not.


    I think it should be obvious that if a function is nonnegative for an entire region, then the area between the function and the x axis can never fall below the x axis.
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    Quote Originally Posted by olski1 View Post
    I have been asked to justify if the following statement is true or false.

    'If the integral of f(x) >= 0 on [a,b], then f(x) is >= 0 on [a,b].

    Would (8-x^2) over the domain [-4,4] be a counter example to this? thus justifying that the statement is false.

    I beleive so as f(x) is <= 0 for some values but the area over that domain is positive. Can anyone tell me if iam on the right path?
    Your counter example is correct, since \int_{-4}^4 8-x^2 ~ dx = 8\int_{-4}^4  dx ~ -  \int_{-4}^4 x^2 ~ dx = 8\cdot 8 - (\frac{(4)^3}{3} - \frac{(-4)^3}{3}) = 64 - \frac{2}{3}64 = \frac{64}{3} > 0

    however f(3) = -1 < 0


    Definitely not.


    I think it should be obvious that if a function is nonnegative for an entire region, then the area between the function and the axis can never fall below the axis.
    I think you were thinking of the other direction of the question.
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