If f(x) takes both positive and negative values,what is geometric intrepretation of Riemann sum?
If 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.
1)If a function is countinous on some closed interval then the integral (Riemann) exists.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.