An integral inequality

In summary, an integral inequality is a mathematical statement that compares the value of an integral to another value. It is unique because it deals with continuous functions and has a wide range of applications in various fields. These inequalities can be solved using different techniques but may have limitations in terms of accuracy and complexity.
  • #1
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Homework Statement


suppose f(x) is monotonely decreasing and positive on [2,+∞),
please compare [∫f(t)dt]^2 and ∫[f(t)]^2dt,
here "∫ "means integrating on the interval [2,x]

Homework Equations


none


The Attempt at a Solution



Maybe the second mean value thereom of integral is helpful.
 
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  • #2
Have you tried anything? In particular, have you selected some simple monotonically decreasing function, such as [itex]f(x)= \frac{1}{x}[/itex] and calculated those two values?
 
  • #3
In fact, yes!
But what I'm really eager to know is how to prove the conclusion.
Maybe when the x is large enough, [∫f(t)dt]^2 is larger.
 

1. What is an integral inequality?

An integral inequality is a mathematical statement that compares the value of an integral (a mathematical concept used to find the area under a curve) to another value. It is typically used to establish relationships between different functions or to prove mathematical theorems.

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

An integral inequality is unique because it deals with the concept of an integral, which is a continuous function, rather than a discrete function. This means that the value of the integral is dependent on the entire shape of the function, rather than just a few specific points.

3. What are some common applications of integral inequalities?

Integral inequalities are used in a variety of fields, including physics, engineering, and economics. They can be used to model real-world situations, solve optimization problems, and prove mathematical theorems.

4. How are integral inequalities solved?

Integral inequalities can be solved using a variety of techniques, including algebraic manipulation, substitution, and integration by parts. The specific approach will depend on the structure of the inequality and the desired outcome.

5. Are there any limitations to using integral inequalities?

While integral inequalities are a powerful tool in mathematics and science, they do have some limitations. For example, they may not always provide the most precise solution to a problem and may require certain assumptions or conditions to be valid. Additionally, solving integral inequalities can be complex and time-consuming.

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