# Real analysis- Riemann Integration.

• bonfire09
In summary: P||]##, then ##r\leq 1-||P||##. This means that ##r\leq x_k## since ##x_k## is the upper bound of the interval ##I_k##. But this also means that ##x_{k-1}\leq r## since ##r\in I_k##. Therefore, we have ##x_{k-1}\leq r\leq x_k##, which is the same as saying that ##r## is in the interval ##I_k##.Finally, you say that ##r\in U_1##, but this is not necessarily true. Remember that ##U_1##
bonfire09

## Homework Statement

Let ##P## be a tagged partition of ##[0,3]##.
Show that the union ##U_1## of all the sub intervals in ##P## with tags in ##[0,1]## satisfies ##[0,1-||P||]\subseteq U_1\subseteq [0,1+||P||]##. (||P|| is the norm of partition P).

## The Attempt at a Solution

We first show that ##[0,1-||P||]\subseteq U_1## with tags in ##[0,1]##. Suppose that ##r\in[0,1-||P||]##. Suppose further ##||P||<1##. It follows that there exists an interval ##I_k =[x_{k-1},x_k]\in P## with tag ##t_k\leq 1## that ##r## belongs to. Since ##||P||<1## then ##x_k-x_{k-1}<1\implies x_k<1+x_{k-1}##. Since ##r\in[0,1-||P||]\implies x_{k-1}\leq r\leq 1-||P||\leq 1\implies x_k<1+x_{k-1}\leq 1+1=2\implies x_k<2## Hence ##r## is in the interval ##I_k## with tag ##t_k\in I_k##. Thus ##r\in U_1##. I think this is wrong and I'm not seeing something here. Any help would be great thanks.

Last edited:

Hello,

Thank you for sharing your attempt at a solution. I think you are on the right track, but there are a few things that need to be clarified and corrected.

First, the statement "We first show that ##[0,1-||P||]\subseteq U_1## with tags in ##[0,1]##" is not entirely accurate. You are trying to show that ##[0,1-||P||]\subseteq U_1## for all tags in ##[0,1]##, not just some tags. This will become clearer in the next steps of your solution.

Second, your proof assumes that ##||P||<1##, but this is not necessarily true. The norm of a partition is defined as the maximum length of the subintervals in the partition, so it can be greater than or equal to 1. Therefore, your proof needs to work for all possible values of ##||P||##.

With these clarifications in mind, let's take a look at your proof:

Suppose that ##r\in[0,1-||P||]## and let ##I_k =[x_{k-1},x_k]\in P## be the interval with tag ##t_k\leq 1## that ##r## belongs to. Here, you are assuming that the tag of the interval is less than or equal to 1, which is not necessarily true. Remember that the tag of an interval is just some point within that interval, so it can be any number between ##x_{k-1}## and ##x_k##. Therefore, you cannot assume that the tag is less than or equal to 1.

To fix this, you can instead say that since ##r\in[0,1-||P||]##, there exists some interval ##I_k =[x_{k-1},x_k]\in P## with tag ##t_k## such that ##r\in I_k##. This is true because the norm of the partition is the maximum length of the subintervals, so there must be at least one subinterval in the partition with a length less than or equal to ##||P||##.

Next, you say that since ##||P||<1##, then ##x_k-x_{k-1}<1##. This is not necessarily true, as mentioned earlier. So instead, you can say that since

## 1. What is Riemann Integration?

Riemann Integration is a mathematical concept that is used to determine the area under a curve in a given interval. It is a method of approximating the area by dividing it into smaller rectangles and summing their areas. This concept is named after the German mathematician Bernhard Riemann.

## 2. What is the difference between Riemann Integration and other methods of integration?

Compared to other methods of integration, such as the Trapezoidal Rule or Simpson's Rule, Riemann Integration provides a more accurate estimation of the area under a curve. This is because it uses a larger number of smaller rectangles to approximate the area, resulting in a more precise calculation.

## 3. How is Riemann Integration used in real-world applications?

Riemann Integration has various applications in fields such as physics, engineering, and economics. In physics, it is used to calculate the work done by a variable force, while in engineering, it is used to determine the amount of material needed to construct a specific shape. In economics, it is used to calculate the total profit or loss of a business over a given period.

## 4. What are the main steps in performing Riemann Integration?

The main steps in Riemann Integration include dividing the given interval into smaller sub-intervals, choosing a sample point within each sub-interval, and calculating the area of each rectangle using the height of the function at the sample point. The final step is to sum the areas of all the rectangles to get an approximation of the total area under the curve.

## 5. Are there any limitations to Riemann Integration?

One limitation of Riemann Integration is that it can only be used for functions that are continuous on the given interval. This means that the function must not have any breaks or jumps in its graph. In addition, Riemann Integration can become more complex when dealing with multi-variable functions, as it requires dividing the given region into smaller sub-regions and applying the integration method to each one separately.

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