If f(x)>= 0 then what is geometrical intrepretation of Riemann sum

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- Nov 14th 2006, 10:04 PMgracyriemann sum
If f(x)>= 0 then what is geometrical intrepretation of Riemann sum

- Nov 14th 2006, 10:06 PMgracyplz need help
If f(x) takes both positive and negative values,what is geometric intrepretation of Riemann sum?

- Nov 14th 2006, 10:14 PMgracyplz help
explain integrability theorem

- Nov 15th 2006, 03:48 AMThePerfectHacker
If $\displaystyle f(x)\geq 0$ 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.

Quote:

explain integrability theorem

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.