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Math Help - riemann sum

  1. #1
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    riemann sum

    If f(x)>= 0 then what is geometrical intrepretation of Riemann sum
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  2. #2
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    plz need help

    If f(x) takes both positive and negative values,what is geometric intrepretation of Riemann sum?
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  3. #3
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    plz help

    explain integrability theorem
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  4. #4
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    Quote Originally Posted by gracy View Post
    If f(x)>= 0 then what is geometrical intrepretation of Riemann sum
    If f(x)\geq 0 and the sum exists (sufficiently a countinous function) then it represents the area below the curve and the x-axis. And if the function gets negative you can think of it as "negative" area.

    explain integrability theorem
    1)If a function is countinous on some closed interval then the integral (Riemann) exists.

    2)The Riemann integral is well-defined by partitioning if it exists. That is no matter how you partition your interval for the Riemann sum it is the same.

    3*)I think the necessary and sufficient condition on 1) is that the function be defined on a closed interval and be discountinous at countably many points. But that definition is often not presented in a Calculus course for it is too advanced. If you can understand it use it.
    Last edited by ThePerfectHacker; November 15th 2006 at 05:05 AM.
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