An integral inequality: Source?

In summary, the conversation is about an integral inequality and its origin. The inequality states that the absolute value of the area under the function f(x) is less than or equal to the area under the absolute value of f(x). The conversation also includes a mention of the Cauchy-Schwarz inequality and its relation to the integral inequality. The participants also discuss the uninteresting nature of this equality and suggest that it is an application of the triangle inequality.
  • #1
philosophking
175
0
I know this is probably a gross generalization of what the actual inequality states, but I'm wondering if someone can tell me the origin of this integral inequality (or something resembling it :/ ):

[tex]|\int f(x)| \leq \int|f(x)|[/tex]

This is my first time using latex, so I hoped that turned out ok. Any suggestions on that too would be appreciated! Thanks.
 
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  • #2
I don't understand your question...

the left side just takes the absolute value of the area under f(x), which could well be negative. As for the right side, it will count a negative f(x) as positive... if it's applied to a velocity function, it would give the total distance traveled rather than displacement for example.
 
  • #3
[tex]|\int f(x)| \leq \int|f(x)|[/tex]

This is not a terribly interesting equality, I am sure that you could form a proof. I can only guess that you are trying to discuss the cauchy-schwarz inequality:

http://mathworld.wolfram.com/SchwarzsInequality.html

Which is reasonably famous but very uninteresting.
 
  • #4
It's just an application of the triangle inequality.
 

1. What is an integral inequality?

An integral inequality is a mathematical statement that compares the value of an integral function with the value of another function. It is often used to prove the existence or non-existence of solutions to certain equations or to establish bounds on the values of certain functions.

2. How is an integral inequality different from other types of inequalities?

An integral inequality involves the integration of a function over a certain interval, whereas other types of inequalities may involve just comparing values of functions at specific points. Integral inequalities also often involve a more complex mathematical setup and may require advanced techniques to solve.

3. What are some applications of integral inequalities?

Integral inequalities have various applications in fields such as physics, engineering, and economics. They can be used to model and analyze real-world situations, such as the rate of change of a physical quantity over time or the growth of a population.

4. How do you prove an integral inequality?

The proof of an integral inequality typically involves using mathematical techniques such as substitution, integration by parts, or comparison with other known functions. It may also require the use of mathematical theorems and properties.

5. Where can I find examples of integral inequalities?

Examples of integral inequalities can be found in textbooks, online resources, and research papers in the field of mathematics. They may also be included in courses on calculus, differential equations, or real analysis.

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