Proving Riemann Integral: Non-Negative f(x)=0 $\rightarrow \int^{b}_{a}f=0$

[danielc]

Homework Statement

Suppose f(x):[a,b]$$\rightarrow$$$$\Re$$ is bounded, non-negative and f(x)=0. Prove that $$\int^{b}_{a}$$f=0.

Homework Equations

The Attempt at a Solution

I am trying to use the idea that lower sums are zero, and show that the upper sums go to zero as the norm of the partition goes to zero. That is Upper sums $$\leq$$ c||P|| such that $$\int^{\overline{b}}_{a}$$f=0.

how can I prove that statement by using the idea above with the choice of c?

 

[matt grime]

Are you saying that f is the zero function? I.e identically zero? (I only ask because you specify that f is bounded and non-negative, which would seem to be redundant.)

Pick any partition. What are the Riemann sums for this partition? (This is trivial to work out, assuming I understand correctly that f(x)=0 for all x in [a,b]).