Integration inequality proof validation

In summary, if f and g are continuous functions with f(x) < g(x) for all x in (a,b), then \int_a^b f(x)\,dx < \int_a^b g(x)\,dx.
  • #1
bonfire09
249
0

Homework Statement


Let ##f:[a,b]\rightarrow\mathbb{R}## and ##g:[a,b]\rightarrow\mathbb{R}## be continuous functions having the property ##f(x)\leq g(x)## for all ##x\in[a,b]##. Prove ##\int_a^b \mathrm f <\int_a^b\mathrm g## iff there exists a point ##x_0## in ##[a,b]## at which ##f(x_0)<g(x_0)##. I just needed help with the reverse direction.


Homework Equations


##R(f,P_n)## stands for the Riemann integral.

The Attempt at a Solution


Suppose there exists a point ##x_0## in ##[a,b]## such that ##f(x_0)<g(x_0)##. Let ##P_n## be a regular partition for ##[a,b]## such that ##\lim_{n\to\infty} ||P_n||=0##. Then there exists an interval ##[x_{j-1},x_j]## such that ##x_0\in[x_{j-1},x_j]##. Let ##c_j=x_0## for some tag ##c_j## where ##1\leq j\leq n##. Picking the same tags for both functions we have ##R(f,P_n)=\sum_{i=1}^{n} f(c_i)(x_i-x_{i-1})=f(c_1)(x_1-x_0)+...+ f(c_j)(x_j-x_{j-1})+...+f(c_n)(x_n-x_{n-1})## and ##R(g,P_n)= \sum_{i=1}^{n} g(c_i)(x_i-x_{i-1})=g(c_1)(x_1-x_0)+...+g(c_j)(x_j-x_{j-1})+..._g(c_n)(x_n-x_{n-1})##. We see by assumption that ##f(c_j)(x_j-x_{j-1})< g(c_j)(x_j-x_{j-1})##. Then ##f(c_1)(x_1-x_0)+...+ f(c_j)(x_j-x_{j-1})+...+f(c_n)(x_n-x_{n-1})< g(c_1)(x_1-x_0)+...+g(c_j)(x_j-x_{j-1})+..._g(c_n)(x_n-x_{n-1})##. Thus ##R(f,P_n)=\sum_{i=1}^{n} f(c_i)(x_i-x_{i-1})<R(g,P_n)= \sum_{i=1}^{n} g(c_i)(x_i-x_{i-1}) \implies R(f,P_n)<R(g,P_n)##. Since ##f## and ##g## are continuous that means they are integrable. Thus we have ##\int_a^b \mathrm f =\lim_{n\to\infty} R(f,P_n)<\lim_{n\to\infty} R(g,P_n)=\int_a^b\mathrm g##

Here is my solution. I need to know if its right or not and how I can clean it up. Thanks.
 
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  • #2
I'm not sure if you have to make it so complicated. In addition, I don't see where you used the properties of continuous to show the inequality for the Riemann sums (you need it!).
Do you know some basic properties about integrals, like the integral of f-g if you know the individual integrals?

(f-g)(x0) < 0, therefore there exists an interval around x0 where ...
 
  • #3
I know about those properties but I thought I would not need those. I thought that since f and g are continuous functions then their both integrable which is what I used to get my last step. Despite my proof being long would it suffice?
 
  • #4
Hmm, I think I understand what you did - picking cj is possible as f and g are continuous, so the justification of this steps comes a bit late. Okay, looks fine, just a bit complicated.
 
  • #5
bonfire09 said:
I know about those properties but I thought I would not need those. I thought that since f and g are continuous functions then their both integrable which is what I used to get my last step. Despite my proof being long would it suffice?

The problem with your proof is that it obscures a key difference between integrable functions in general and continuous functions in particular. To that extent it doesn't promote understanding.

It is true for all integrable functions that if [itex]f(x) < g(x)[/itex] everywhere in [itex](a,b)[/itex] then
[tex]
\int_a^b f(x)\,dx < \int_a^b g(x)\,dx.
[/tex]

But if [itex]f[/itex] and [itex]g[/itex] are integrable and such that [itex]f(x) = g(x)[/itex] except at a finite number of points, then [tex]\int_a^b f(x)\,dx = \int_a^b g(x)\,dx.[/tex]
So the restriction to continuous [itex]f[/itex] and [itex]g[/itex] must somehow exclude this case, and that's what you should focus on.

Also, proofs involving Riemann sums take considerable time to write out in examinations, so shorter proofs using basic properties of the integral are preferable where possible. (If you want practice in manipulating upper and lower sums, try proving that the second fact I asserted is indeed true.)
 

Related to Integration inequality proof validation

What is integration inequality proof validation?

Integration inequality proof validation is the process of checking the validity and accuracy of a mathematical proof that involves integration inequalities. This is done by examining each step of the proof and making sure that it follows logical reasoning and mathematical rules.

Why is it important to validate integration inequality proofs?

Validating integration inequality proofs is important because it ensures the accuracy and reliability of mathematical results. It also helps to prevent errors and mistakes in calculations, which can have significant consequences in various fields such as physics, engineering, and economics.

What are the common techniques used for integration inequality proof validation?

Some common techniques used for integration inequality proof validation include checking the domain of the function, using intermediate value theorem, and using the first and second derivative tests. Other techniques may involve substitution, trigonometric identities, and integration by parts.

What are some challenges faced in validating integration inequality proofs?

One of the main challenges in validating integration inequality proofs is the complexity of the proofs. These proofs can involve multiple steps and require a deep understanding of mathematical concepts. Another challenge is that small errors in calculations can lead to incorrect results, making it important to be thorough in the validation process.

How can one improve their skills in integration inequality proof validation?

To improve skills in integration inequality proof validation, one can practice solving various integration inequality problems and familiarize themselves with different proof techniques. It is also helpful to seek guidance from experienced mathematicians and to continuously learn and review mathematical concepts.

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