Math Help - riemann sum

1. riemann sum

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

2. plz need help

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

3. plz help

explain integrability theorem

4. Originally Posted by gracy
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.