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

• danielc
In summary, the conversation discusses the proof of the statement that if f(x) is a bounded, non-negative function on the interval [a,b] and f(x)=0, then the integral of f(x) from a to b is equal to 0. The solution involves using the idea that lower sums are zero and showing that the upper sums go to zero as the partition norm approaches zero. The conversation also mentions considering the Riemann sums for any partition to prove this statement.
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.

## 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?

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]).

## 1. What is the Riemann Integral and why is it important?

The Riemann Integral is a method of calculating the area under a curve on a given interval. It is important because it allows us to find the exact value of an integral, which has many applications in mathematics, physics, and engineering.

## 2. Can you explain the concept of non-negativity in relation to the Riemann Integral?

In the context of the Riemann Integral, non-negativity means that the function being integrated must have non-negative values (greater than or equal to zero) on the given interval. This ensures that the area under the curve is always positive and can be accurately calculated.

## 3. How does the fact that f(x)=0 on the interval affect the value of the integral?

If f(x)=0 on the interval, then the integral will also be equal to 0. This is because the area under the curve is 0 since the function has no height (or value) above the x-axis.

## 4. Can you prove that if f(x)=0 on the interval, then the integral is equal to 0?

Yes, the proof involves using the definition of the Riemann Integral and the properties of integration to show that when f(x)=0, the integral will always be equal to 0. This proof is a fundamental part of understanding the Riemann Integral.

## 5. Are there any exceptions to the rule that f(x)=0 $\rightarrow \int^{b}_{a}f=0$?

Yes, there are some exceptions to this rule. For example, if the function is not defined on the interval or if it has discontinuities, the value of the integral may not be equal to 0 even if f(x)=0 on the interval. Additionally, if the function is not integrable, the rule may not apply. However, in general, this rule is true and has been proven to hold for most functions.

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